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Chapter 12: Problem 75

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$\log _{4} x=-3$$

Short Answer

Expert verified
The equivalent exponential form of the equation is \(4^{-3} = x\). The solution for \(x\) is \(x = \frac{1}{64}\).

Step by step solution

01

Transform Logarithmic Equation to Exponential Form

Given the equation \(\log_{4}x = -3\), you can use the rule that \(\log_b a = c \) is equivalent to \( b^c = a \). By applying this rule, the equation \(\log _{4} x=-3\) is equivalent to \(4^{-3} = x\).
02

Solve for \(x\)

To find the value of \(x\), you can simplify \(4^{-3}\). This is equivalent to \(\frac{1}{4^3}\), which simplifies further to \(\frac{1}{64}\). Therefore, \(x = \frac{1}{64}\).

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Most popular questions from this chapter

Logarithmic models are well suited to phenomena in which growth is initially rapid, but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

What question can be asked to help evaluate \(\log _{3} 81 ?\)

What is a logarithmic equation?
See all solutions

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