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Chapter 3: Problem 72

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$5^{x}=212$$

Short Answer

Expert verified
The value of x is given by the formula \(x = \frac{\ln(212)}{\ln(5)}\). Use a graphing utility to find the approximate value up to three decimal places. Then verify the obtained solution by substitution in the original equation.

Step by step solution

01

Taking Logarithm on Both Sides

Take natural logarithm (ln) on both sides of the equation:\[\ln(5^{x}) = \ln(212)\] because of the rule \(\ln(a^{b}) = b \cdot \ln(a)\), this can simplify to \(x \cdot \ln(5) = \ln(212)\)
02

Solve for x

Find the value of x by just dividing the ln(212) by ln(5):\[x = \frac{\ln(212)}{\ln(5)}\]
03

Use a Graphing Utility and Approximate Result

Plot the equation in a graphing utility. The x-coordinate of the intersection point of y=\(\ln(5^{x})\) and y=\(\ln(212)\) will be the solution to the equation. Note down the approximate x-value of this intersection point up to three decimal places.
04

Verify the Result Algebraically

Substitute the obtained value of x in the original equation, and verify if the left-hand side equals the right-hand side. If yes, the solution is correct.

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