Solve (5/3) ^ - 5 * (5/3) ^ - 3 = (5/3) ^ (-2x) | 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

$$(\frac{5}{3})^{-5}\times(\frac{5}{3})^{-3}=(\frac{5}{3})^{-2x}$$

Solve for x

$x=4$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add $-5$ and $-3$ to get $-8$.

$$\left(\frac{5}{3}\right)^{-8}=\left(\frac{5}{3}\right)^{-2x}$$

Calculate $\frac{5}{3}$ to the power of $-8$ and get $\frac{6561}{390625}$.

$$\frac{6561}{390625}=\left(\frac{5}{3}\right)^{-2x}$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{5}{3}\right)^{-2x}=\frac{6561}{390625}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{3}\right)^{-2x})=\log(\frac{6561}{390625})$$

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

$$-2x\log(\frac{5}{3})=\log(\frac{6561}{390625})$$

Divide both sides by $\log(\frac{5}{3})$.

$$-2x=\frac{\log(\frac{6561}{390625})}{\log(\frac{5}{3})}$$

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

$$-2x=\log_{\frac{5}{3}}\left(\frac{6561}{390625}\right)$$

Divide both sides by $-2$.

$$x=-\frac{8}{-2}$$

Solve for x (complex solution)

$x=\frac{\pi n_{1}i}{\ln(\frac{3}{5})}-\log_{\frac{3}{5}}\left(\frac{625}{81}\right)$
$n_{1}\in \mathrm{Z}$