/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 56

Solve the following equations. $$5^{3 x}=29$$

Short Answer

Expert verified
Question: Solve the exponential equation: $$5^{3x} = 29$$, and find the value of x. Answer: $$x \approx 0.748$$

Step by step solution

01

Write the equation

The given equation is: $$5^{3 x}=29$$
02

Applying Logarithms

To solve this exponential equation, we will use logarithms. Take the logarithm on both sides of the equation. It is often easier to work with the natural logarithm (ln) since it has a simpler base, so we will take the ln of both sides: $$\ln(5^{3 x})=\ln(29)$$
03

Simplify and Solve

Now use the logarithmic property \(\ln(a^b)=b\ln(a)\) to simplify the left side of the equation: $$3x \cdot \ln(5)=\ln(29)$$ Next, isolate x by dividing both sides of the equation by \(3\ln(5)\): $$x = \frac{\ln(29)}{3\ln(5)}$$
04

Calculate the Value of x

Using a calculator, find the value of x: $$x = \frac{\ln(29)}{3\ln(5)} \approx 0.748$$
05

Verify the Solution

To verify the solution, substitute the value of x into the original equation and check if the left-hand side is equal to the right-hand side: $$5^{3(0.748)} \approx 29$$ Since both sides are approximately equal, the solution is correct. Therefore, the answer to the given equation is: $$x \approx 0.748$$

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