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Chapter 12: Problem 15

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$6^{\frac{x-3}{4}}=\sqrt{6}$$

Short Answer

Expert verified
The solution to the equation is \(x = 5\)

Step by step solution

01

Setting up the equation

Start by writing down the equation. The equation given is \(6^{\frac{x-3}{4}}=\sqrt{6}\).
02

Express both sides using the same base

The next step is to convert the right side of the equation to have the same base as the left side, i.e., base 6. To do this, remember that the square root function is equivalent to raising a number to the power of a half, i.e. \(\sqrt{a} = a^{0.5}\). In this case, \(\sqrt{6}\) can be rewritten as \(6^{0.5}\).
03

Equating the exponents

Since we have both sides of the equation expressed with same base now, we can equate the expressions on the exponents of both sides of the equation. As for as our equation is concerned, the expressions in the exponents are \(\frac{x-3}{4}\) and 0.5 (or \(\frac{1}{2}\)). So, we have the equation \(\frac{x-3}{4} = 0.5\)
04

Solving for x

The next step is to solve for x. To do this, the first step is to get rid of the fraction by multiplying both sides of the equation by 4, giving \(x-3 = 2\). Then, we add 3 to both sides of the equation to isolate x, giving \(x = 2 + 3\)
05

Answer

From the previous step, we have \(x = 2 + 3\). Adding 2 and 3 gives \(x = 5\)

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Most popular questions from this chapter

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. $$2^{x+1}=8$$

What is a logarithmic equation?

Describe the change-of-base property and give an example.

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the \(\operatorname{graph}\) of \(g(x)=\log _{2} x\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)
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