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Chapter 4: Problem 8

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{3 x-1}=125$$

Short Answer

Expert verified
The solution to the exponential equation \(5^{3x-1}=125\) is \(x = 4/3\).

Step by step solution

01

Express the Right Side as a Power of 5

Observe that 125 is a power of 5, in fact 125 is \(5^3\). So, replace 125 with \(5^3\) in the equation. Now, the original equation \(5^{3x-1}=125\) becomes \(5^{3x-1}=5^3\)
02

Equate the Exponents

Since the bases on both sides of the equation are now the same (base 5), the exponents must also be equal. So, set the exponents equal to each other: \(3x - 1 = 3\)
03

Solve for 'x'

In order to solve for 'x', first add 1 to both sides to isolate '3x' on the left side of the equation. This gives: \(3x = 3 + 1\), or \(3x = 4\). Finally, divide both sides by 3 to get: \(x = 4/3\).

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Most popular questions from this chapter

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=4.5(0.6)^{x} $$

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