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.

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

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

Note

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.

Example 4

Evaluate.

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

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

Note

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

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