Suggested languages for you:

Americas

Europe

Q41.

Expert-verified
Found in: Page 352

### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

# Logarithmic Equations Use the definition of the logarithmic function to find x.(a) ${\mathrm{log}}_{2}\left(\frac{1}{2}\right)=x$ (b) ${\mathrm{log}}_{10}x=-3$

1. The required value of x is $-1$.
2. The required value of x is 0.001.
See the step by step solution

## Part a. Step 1. Given.

The given equation is ${\mathrm{log}}_{2}\left(\frac{1}{2}\right)=x$.

## Part a. Step 2. To determine.

We have to find the value of x using the definition of the logarithmic function.

## Part a. Step 3. Calculation.

We’ll use the definition of the logarithmic function ${\mathrm{log}}_{a}c=b\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{a}^{b}=c$.

Comparing ${\mathrm{log}}_{2}\left(\frac{1}{2}\right)=x$ with ${\mathrm{log}}_{a}c=b\text{\hspace{0.17em}}$ we get $a=2,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}b=x,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}c=\frac{1}{2}$.

So, the equivalent exponential form is:

${a}^{b}=c$

or, ${2}^{x}=\frac{1}{2}$

or, ${2}^{x}={2}^{-1}$

or, $x=-1$ [Equated the exponents, since the bases are same]

Hence, the required value of x is $-1$.

## Part b. Step 1. Given.

The given equation is ${\mathrm{log}}_{10}x=-3$.

## Part b. Step 2. To determine.

We have to find the value of x using the definition of the logarithmic function.

## Part b. Step 3. Calculation.

We’ll use the definition of the logarithmic function ${\mathrm{log}}_{a}c=b\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{a}^{b}=c$.

Comparing ${\mathrm{log}}_{10}x=-3$ with ${\mathrm{log}}_{a}c=b$ we get $a=10,\text{\hspace{0.17em}\hspace{0.17em}}b=-3,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}c=x$.

So, the equivalent exponential form is:

${a}^{b}=c$

or, ${10}^{-3}=x$

or, $\frac{1}{1000}=x$

or, $0.001=x$

Hence, the required value of x is 0.001.