3.5: Properties of Logarithms

Learning Objectives

1. Evaluate Logarithms Whose base is Neither 10 Nor e.
2. Write a Logarithmic Expression as a Sun of Difference of Logarithms (Expand logarithms).
3. Write a Logarithmic Expression as a Single Logarithm (Condense logarithms).

Change-of-Base Formula

[latex]If\;a,\;b,\;x\in\mathbb{R}\;and\;a,\;b,\;x>0\;and\;a,\;b\;\neq1,[/latex]

[latex]\log_b\left(x\right)=\frac{\log_a\left(x\right)}{\log_a\left(b\right)}=\frac{\log\left(x\right)}{\log\left(b\right)}=\frac{\ln\left(x\right)}{\ln\left(b\right)}[/latex]

If the logarithmic base is neither 10 nor e, we need to convert it to a common base, such as 10 or e, to evaluate the logarithm.

Example 3.5-1-1: Use the Change-of-Base Formula and a calculator to evaluate the logarithm, round it to nearest hundredth.

[latex]\log_{1.5}\left(38\right)[/latex]

Key

Example 3.5-1-1: Use the Change-of-Base Formula and a calculator to evaluate the logarithm, round it to nearest hundredth.

[latex]\log_{1.5}\left(38\right)[/latex]

[latex]\log_b\left(x\right)=\frac{\log_a\left(x\right)}{\log_a\left(b\right)}=\frac{\log\left(x\right)}{\log\left(b\right)}=\frac{\ln\left(x\right)}{\ln\left(b\right)}[/latex]

Based on the formula, we can select either the natural log or the common log or log with a same base to change the base of the log in the original question. Remember, you only need to use one of them. If use natural log: [latex]\log_b\left(x\right)=\frac{\ln\left(x\right)}{\ln\left(b\right)}[/latex]

We have

[latex]{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol x}{\color[rgb]{0.5, 0.0, 0.5}\mathbf\rightarrow}{\color[rgb]{0.5, 0.0, 0.5}\mathbf38}[/latex]

[latex]{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol b}{\color[rgb]{0.0, 0.0, 1.0}\mathbf\rightarrow}{\color[rgb]{0.0, 0.0, 1.0}\mathbf1}{\color[rgb]{0.0, 0.0, 1.0}\mathbf.}{\color[rgb]{0.0, 0.0, 1.0}\mathbf5}[/latex]

[latex]\log_{{\color[rgb]{0.0, 0.0, 1.0}\mathbf1}{\color[rgb]{0.0, 0.0, 1.0}\mathbf.}{\color[rgb]{0.0, 0.0, 1.0}\mathbf5}}\left({\color[rgb]{0.5, 0.0, 0.5}\mathbf38}\right)=\frac{\ln\left(38\right)}{\ln\left(1.5\right)}=8.9713913404...\approx8.97[/latex]

Product Rule for Logarithms

Given any real number [latex]x[/latex] and positive real numbers [latex]M, N,[/latex] and [latex]b[/latex], where [latex]b\neq1[/latex], we will show

[latex]\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)[/latex]

Let [latex]m=\log_bM[/latex] and [latex]n=\log_bN[/latex]. In exponential form, these equations are [latex]b^m=M[/latex] and [latex]b^n=N[/latex]. It follows that

[latex]\log_b\left(M\cdot N\right)=\log_b\left(M\right)+\log_b\left(N\right)[/latex]

Example 3.5-2-1: Using the product rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(3x\right)[/latex]

Key

Example 3.5-2-1: Using the product rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(3x\right)[/latex]

Since: [latex]\log_b\left(M\cdot N\right)=\log_b\left(M\right)+\log_b\left(N\right)[/latex]

[latex]\log_b\left(3x\right)=\log_b\left(3\right)+\log_b\left(x\right)[/latex]

Quotient Rule for Logarithms

Given any real number [latex]x[/latex] and positive real numbers [latex]M, N,[/latex] and [latex]b[/latex], where [latex]b\neq1[/latex], we will show

[latex]\log_b\left(\frac MN\right)=\log_b\left(M\right)-\log_b\left(N\right)[/latex]

Let [latex]m=\log_bM[/latex] and [latex]n=\log_bN[/latex]. In exponential form, these equations are [latex]b^m=M[/latex] and [latex]b^n=N[/latex]. It follows that

[latex]\log_b\left(\frac MN\right)=\log_b\left(M\right)-\log_b\left(N\right)[/latex]

Example 3.5-2-2: Using the quotient rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(\frac3x\right)[/latex]

Key

Example 3.5-2-2: Using the quotient rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(\frac3x\right)[/latex]

Since: [latex]\log_b\left(\frac MN\right)=\log_b\left(M\right)-\log_b\left(N\right)[/latex]

[latex]\log_b\left(\frac3x\right)=\log_b\left(3\right)-\log_b\left(x\right)[/latex]

Power Rule for Logarithms

We’ve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as [latex]x^2[/latex] ? One method is as follows:

[latex]\log_b\left(x^2\right)=\log_b\left(x\cdot x\right)[/latex]

[latex]=\log_b\left(x\right)+\log_b\left(x\right)[/latex]

[latex]=2\log_bx[/latex]

[latex]\log_b\left(x^r\right)=r\cdot\log_b\left(x\right)[/latex]

Example 3.5-2-3: Using the power rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(x^8\right)[/latex]

Key

Example 3.5-2-3: Using the power rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(x^8\right)[/latex]

Since: [latex]\log_b\left(x^r\right)=r\log_b\left(x\right)[/latex]

[latex]\log_b\left(x^8\right)=8\log_b\left(x\right)[/latex]

Steps to expand

Expand:

1. quotient (change to “[latex]-[/latex]”)
2. product (change to “[latex]+[/latex]”)
3. power (bring the power down)

Example 3.5-2-4: Using the product, quotient, and power rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(\frac{3x^8}{5yz}\right)[/latex]

Key

Example 3.5-2-4: Using the product, quotient, and power rules for logarithms, expand the logarithmic expressions.

To check if it’s fully expanded:

1. No fractions,
2. No multiplication,
3. No exponents.

[latex]\log_b\left(\frac{3x^8}{5yz}\right)[/latex]

1. quotient (change to “[latex]-[/latex]”)

[latex]\log_b\left(\frac{3x^8}{5yz}\right)=\log_b\left(3x^8\right)-\log_b\left(5yz\right)[/latex]

2. product (change to “[latex]+[/latex]”)

[latex]\log_b\left(3x^8\right)-\log_b\left(5yz\right)=\left(\log_b\left(3\right)+\log_b\left(x^8\right)\right)-\left(\log_b\left(5\right)+\log_b\left(y\right)+\log_b\left(z\right)\right)[/latex]

3. power (bring the power down)

[latex]\begin{array}{c}\left(\log_b\left(3\right)+\log_b\left(x^{\color[rgb]{0.0, 0.0, 1.0}\mathbf8}\right)\right)-\left(\log_b\left(5\right)+\log_b\left(y\right)+\log_b\left(z\right)\right)\\=\left(\log_b\left(3\right)+{\color[rgb]{0.0, 0.0, 1.0}\mathbf8}\log_b\left(x\right)\right)-\left(\log_b\left(5\right)+\log_b\left(y\right)+\log_b\left(z\right)\right)\end{array}[/latex]

Last step: simplify

[latex]\begin{array}{c}\left(\log_b\left(3\right)+8\log_b\left(x\right)\right){\color[rgb]{0.5, 0.0, 0.5}\mathbf-}\left(\log_b\left(5\right)+\log_b\left(y\right)+\log_b\left(z\right)\right)\\=\log_b\left(3\right)+8\log_b\left(x\right){\color[rgb]{0.5, 0.0, 0.5}\mathbf-}\log_b\left(5\right){\color[rgb]{0.5, 0.0, 0.5}\mathbf-}\log_b\left(y\right){\color[rgb]{0.5, 0.0, 0.5}\mathbf-}\log_b\left(z\right)\end{array}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol d}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol i}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol s}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol t}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol r}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol i}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol b}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol u}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol t}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol e}{\color[rgb]{0.5, 0.0, 0.5}\mathbf\;}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol t}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol h}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol e}{\color[rgb]{0.5, 0.0, 0.5}\mathbf\;}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol n}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol e}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol g}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol a}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol t}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol i}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol v}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol e}{\color[rgb]{0.5, 0.0, 0.5}\mathbf\;}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol s}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol i}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol g}{\color[rgb]{0.5, 0.0, 0.5}\boldsymbol n}[/latex]

Answer:

[latex]\log_b\left(3\right)+8\log_b\left(x\right)-\log_b\left(5\right)-\log_b\left(y\right)-\log_b\left(z\right)[/latex]

Condense

Condensing means combining logarithms. It is the reverse process of expanding. We apply the rules in this order: power, quotient, then product.

• [latex]\log_b\left(\frac xy\right)=\log_b\left(x\right)-\log_b\left(y\right)[/latex]
• [latex]\log_b\left(x\cdot y\right)=\log_b\left(x\right)+\log_b\left(y\right)[/latex]
• [latex]\log_b\left(x^r\right)=r\cdot\log_b\left(x\right)[/latex]

The rules above can be applied from left to right to expand, or from right to left to condense.

Steps:

1. power (coefficient)
2. product (“[latex]+[/latex]”)
3. quotient (“[latex]-[/latex]”)

Example 3.5-3-1: Using the product, quotient, and power rules for logarithms, condense the logarithmic expressions.

[latex]\log_b\left(x\right)+8\log_b\left(y\right)-\log_b\left(7\right)-3\log_b\left(z\right)[/latex]

Key

Example 3.5-3-1: Using the product, quotient, and power rules for logarithms, condense the logarithmic expressions.

[latex]\log_b\left(x\right)+8\log_b\left(y\right)-\log_b\left(7\right)-3\log_b\left(z\right)[/latex]

1. power (coefficient)

[latex]\log_b\left(x\right)+{\color[rgb]{0.5, 0.0, 0.5}\mathbf8}\log_b\left(y\right)-\log_b\left(7\right)-{\color[rgb]{0.5, 0.0, 0.5}\mathbf3}\log_b\left(z\right)[/latex]

[latex]\log_b\left(x\right)+\log_b\left(y^{\color[rgb]{0.5, 0.0, 0.5}\mathbf8}\right)-\log_b\left(7\right)-\log_b\left(z^{\color[rgb]{0.5, 0.0, 0.5}\mathbf3}\right)[/latex]

2. product (“[latex]+[/latex]”)

[latex]\log_b\left(xy^8\right)-\log_b\left(7\right)-\log_b\left(z^{\color[rgb]{0.5, 0.0, 0.5}\mathbf3}\right)[/latex]

3. quotient (“[latex]-[/latex]”)

All negative logarithmic terms are in the denominator

[latex]\log_b\left(xy^8\right){\color[rgb]{0.0, 0.0, 1.0}\mathbf-}{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol l}{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol o}{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol g}_{\color[rgb]{0.0, 0.0, 1.0}\mathbf b}\mathbf{\color[rgb]{0.0, 0.0, 1.0}\left(7\right)}{\color[rgb]{0.0, 0.0, 1.0}\mathbf-}{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol l}{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol o}{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol g}_{\color[rgb]{0.0, 0.0, 1.0}\mathbf b}\mathbf{\color[rgb]{0.0, 0.0, 1.0}\left(z^3\right)}[/latex]

[latex]=\log_b\left(\frac{xy^8}{{\color[rgb]{0.0, 0.0, 1.0}\mathbf7}{\color[rgb]{0.0, 0.0, 1.0}\boldsymbol z}^{\color[rgb]{0.0, 0.0, 1.0}\mathbf3}}\right)[/latex]

Answer: [latex]\log_b\left(\frac{xy^8}{7z^3}\right)[/latex]