Solve ( ii ) 5 ^ x = 1/125 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$5 ^ { x } = \frac { 1 } { 125 }$$

Solve for x

$x=-3$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$5^{x}=\frac{1}{125}$$

Take the logarithm of both sides of the equation.

$$\log(5^{x})=\log(\frac{1}{125})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$x\log(5)=\log(\frac{1}{125})$$

Divide both sides by $\log(5)$.

$$x=\frac{\log(\frac{1}{125})}{\log(5)}$$

By the change-of-base formula $\frac{\log(a)}{\log(b)}=\log_{b}\left(a\right)$.

$$x=\log_{5}\left(\frac{1}{125}\right)$$

Solve for x (complex solution)

$x=\frac{2\pi n_{1}i}{\ln(5)}-3$
$n_{1}\in \mathrm{Z}$