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Chapter 3: Problem 33

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$6\left(2^{3 x-1}\right)-7=9$$

Short Answer

Expert verified
The solution to the given exponential equation, approximated to three decimal places, is \(x \approx 0.356\).

Step by step solution

01

Isolate the Exponential Expression

The first step involves simplifying the equation in order to isolate the exponential term. So, start by adding 7 to both sides of the equation: \(6(2^{3x-1})=16\). Next, divide both sides by 6 to completely isolate the exponential term: \(2^{3x-1}=2.67\).
02

Apply Natural Logarithm to both sides

In this step, apply the natural logarithm (ln) to both sides of the equation to remove the exponent. This gives: \(ln(2^{3x-1})=ln(2.67)\) . Using the logarithmic property that allows us to move the exponent in front, the equation becomes: \((3x-1)ln(2)=ln(2.67)\).
03

Solve for x

Now, the equation can be simplified further to solve for x. Firstly, distribute ln(2) to get: \(3xln(2)-ln(2)=ln(2.67)\). Then, add ln(2) to both sides: \(3xln(2)=ln(2.67)+ln(2)\). Finally, divide by 3ln(2) to solve for x: \(x=\frac{ln(2.67)+ln(2)}{3ln(2)}\).
04

Approximate the Result

The last step is to calculate the numerical value of x, approximated to three decimal places. Using a calculator to compute the natural log values and the division, we get x approximately equal to 0.356.

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Most popular questions from this chapter

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