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Chapter 4: Problem 18

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$9^{x}=\frac{1}{\sqrt[3]{3}}$$

Short Answer

Expert verified
The solution to the equation is \(x = -1/6\)

Step by step solution

01

Expressing the base

First, rewrite both sides of the equation using the same base. The number 9 is actually \(3^2\) and the number 1 over cube root of 3 is \((3^{-1/3})\). So, we rewrite the equation as \((3^2)^x = 3^{-1/3}\). This simplifies to \(3^{2x} = 3^{-1/3}\).
02

Equating Exponents

Since the bases are the same on both sides of the equation, we can set the exponents equal to each other. Thus, \(2x = -1/3\).
03

Solving for x

Next, solve the equation for x by dividing both sides by 2. Therefore, \(x = -1/6\).

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Most popular questions from this chapter

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

In many states, a \(17 \%\) risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do you agree with the \(17 \%\) risk as a cutoff percentage, or do you feel that the percentage should be lower or higher? Explain your answer. What blood alcohol concentration corresponds to what you believe is an appropriate percentage?

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$3^{x}=2 x+3$$

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.
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