Explore: Evaluating Logarithmic Expressions Solutions

Evaluating Logarithmic Expressions Solutions

Foundational properties of logs when a>0, a≠1 :  
- If $a^{0} = 1$ , then $\log_{a} 1 = 0$  
- If $a^{n} = a^{n}$ , then $\log_{a} (a^{n}) = n$  
Foundational log properties and exponent rules are used to evaluate logs without using technology. 
A common base must exist to find the value of the variable.

To evaluate logarithms (by hand): 

Set the expression equal to a variable . (In this lesson, use x.) 
Convert the logarithmic equation to an exponential equation. 
Write both sides of the equation with the same base . 
Solve .

Note: When no base is given, the base, b, is 10.

Example 3

Evaluate.

Note

Problems A–B

When looking at both problems, notice that the x value can be negative or a fraction. It is important to carefully write the equation in exponential form so you can determine the correct value.

$\log_{16} 2$

$\begin{array}{rcl} \log_{16} 2 & = & x \\ 16^{x} & = & 2 \\ 2^{4 x} & = & 2^{1} \\ 4 x & = & 1 \\ x & = & \frac{1}{4} \\ \log_{16} 2 & = & \frac{1}{4} \end{array}$

$\log_{2} (\frac{1}{16})$

$\begin{array}{rcl} \log_{2} (\frac{1}{16}) & = & x \\ 2^{x} & = & \frac{1}{16} \\ 2^{x} & = & 2^{- 4} \\ x & = & - 4 \end{array}$

Example 4

Evaluate.

Note

Problems A–B

While b cannot equal 1 and must be greater than zero, problems can still have decimals, fractions, and negative and positive answers, etc.

$\log_{0.25} 64$

$\begin{array}{rcl} \log_{0.25} 64 & = & x \\ {(\frac{1}{4})}^{x} & = & 64 \\ 4^{- 1 x} & = & 4^{3} \\ - 1 x & = & 3 \\ x & = & - 3 \end{array}$

$\log_{3} (\sqrt[5]{3})$

$\begin{array}{rcl} \log_{3} (\sqrt[5]{3}) & = & x \\ 3^{x} & = & \sqrt[5]{3} \\ 3^{x} & = & 3^{\frac{1}{5}} \\ x & = & \frac{1}{5} \end{array}$

Most logs are irrational numbers, and therefore, technology is used to estimate the value of a logarithm. 

As with evaluating logarithms by hand, when no base is given, the base, b, is 10, which is the default base for calculators.

Example 5

Evaluate with a calculator to the ten thousandths (four decimal places).

$\log 8$

0.903089987 = 0.9031

$\log 225$

2.352182518 = 2.3522