The exponential and logarithmic functions introduction page focused on both types of functions and the relationship between them.
Here we will look at some of the types of exponential and logarithmic functions examples and problems that can be encountered, and how they can be solved.
Examples
(1.1)
Write {\tt{log}}_4 (64) \space {\small{=}} \space 3 in exponential form.
Solution
{\tt{log}}_4 (64) \space {\small{=}} \space 3 => 4^3 \space {\small{=}} \space 64
(1.2)
Write 2^4 \space {\small{=}} \space 16 in exponential form.
Solution
2^4 \space {\small{=}} \space 16 => {\tt{log}}_2 (16) \space {\small{=}} \space 4
(1.3)
Express 5^2 \space {\small{=}} \space t in logarithmic form.
Solution
5^2 \space {\small{=}} \space t => {\tt{log}}_5 (t) \space {\small{=}} \space 2
(1.4)
What value is t where {\tt{log}}_5 (t) \space {\small{=}} \space 3 ?
Solution
{\tt{log}}_5 (t) \space {\small{=}} \space 3 => 5^3 \space {\small{=}} \space t => 125 \space {\small{=}} \space t
(1.5)
Evaluate {\tt{log}}_a (\frac{1}{32}) \space {\small{=}} \space {\text{-}}5.
Solution
a^{{\text{-}}5} \space {\small{=}} \space \frac{1}{32} => \frac{1}{a^5} \space {\small{=}} \space \frac{1}{32} => a \space {\small{=}} \space 2
Exponential and Logarithmic Functions
Examples, Further
(2.1)
Solve for x where {\tt{log}}_x (9) \space {\small{=}} \space 4.
Solution
{\tt{log}}_x (9) \space {\small{=}} \space 4 => x^4 \space {\small{=}} \space 9
x^4 \space {\small{=}} \space 9
(x^2)^2 \space {\small{=}} \space 9
x^2 \space {\small{=}} \space 3 , x \space {\small{=}} \space \sqrt{3}
(2.2)
Solve for r where {\tt{log}}_{625} (5) \space {\small{=}} \space r.
Solution
{\tt{log}}_{625} (5) \space {\small{=}} \space r => 625^r \space {\small{=}} \space 5
At this point, it helps to know that,
{\tt{log}}_5 (625) \space {\small{=}} \space 4 => 5^4 \space {\small{=}} \space 625.
So we can write 625^r \space {\small{=}} \space 5 as (5^4)^r \space {\small{=}} \space 5.
Now:
(5^4)^r \space {\small{=}} \space 5^1 => 5^{4r} \space {\small{=}} \space 5^1 => 4r \space {\small{=}} \space 1 , r \space {\small{=}} \space \frac{1}{4}
(2.3)
Solve for p where {\tt{log}}_{0.5} (p) \space {\small{=}} \space {\text{-}}4.
Solution
{\tt{log}}_{0.5} (p) \space {\small{=}} \space {\text{-}}4 => 0.5^{{\text{-}}4} \space {\small{=}} \space p
0.5^{{\text{-}}4} \space {\small{=}} \space p => (\frac{1}{2})^{{\text{-}}4} \space {\small{=}} \space p
Now:
(\frac{2}{1})^4 \space {\small{=}} \space p => 16 \space {\small{=}} \space p