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Chapter 1: Problem 42

Solve the following equations. $$\log _{5} x=-1$$

Short Answer

Expert verified
Question: Solve the logarithmic equation: $$\log _{5} x=-1$$. Answer: $$x=\frac{1}{5}$$

Step by step solution

01

Identify the base, exponent, and result

In the given equation $$\log _{5} x = -1$$, the base is 5, the exponent is -1, and the result is x.
02

Transform the logarithmic equation into an exponential equation

Using the property mentioned in the analysis, we can rewrite the equation as $$5^{-1} = x$$.
03

Simplify the exponential equation

Recall that any number raised to a negative exponent equals the reciprocal of that number raised to the positive exponent. So $$5^{-1} = \frac{1}{5^1} = \frac{1}{5}$$ which gives us $$\frac{1}{5} = x$$.
04

Write the final solution

The solution to the given equation $$\log _{5} x=-1$$ is $$x = \frac{1}{5}$$.

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