Question

What is the relationship between exponentials and logarithms? How can you use these to solve equations? Provide an example in your explanation.

Answer 1

Answer:

Exponentials and logarithms are inverses of each other.

Step-by-step explanation:

Exponentials and logarithms are inverses of each other.

For logarithmic function:

Domain = $\left ( 0,\infty \right )$, Range = $\left ( -\infty ,\infty \right )$

Vertical asymptote is y - axis.

x - intercept is (1,0)

For exponential function:

Domain = $\left ( -\infty ,\infty \right )$, Range = $\left ( 0,\infty \right )$

Horizontal asymptote is x - axis.

y- intercept is (0,1)

Both exponential and logarithmic functions are increasing.

For example:

Solve: $\log x=\frac{\log 5+\log 3}{\log 3^2}$

$\log x=\frac{\log 5+\log 3}{\log 3^2}\\\log x=\frac{\log (5\times 3)}{2\log 3}\,\,\left \{ \because \log (ab)=\log a+\log b\,,\,\log a^b=b\log a \right \}\\=\frac{\log 15}{2\log 3}$

$\Rightarrow \log x=\frac{\log 15}{2\log 3}\\\Rightarrow x=e^{\frac{\log 15}{2\log 3}}\,\,\left \{ \because \log x=y\Rightarrow x=e^y \right \}$