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Chapter 3: Problem 57

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 equation is approximately \(x=0.433\).

Step by step solution

01

Isolate the Exponential Expression

First, add 7 to both sides of the equation to isolate the exponential expression on one side of the equation. The equation then becomes: \(6(2^{3x-1})=16\). Then divide both sides of the equation by 6 to fully isolate the exponential expression: \(2^{3x-1}=16/6\).
02

Convert to Logarithmic Form

Convert the equation from exponential form to logarithmic form. The equation then becomes: \( \log_{2} (16/6)= 3x-1\).
03

Simplify the Logarithmic Expression

Simplify the logarithmic equation by evaluating \(\log_{2} (16/6)\) to get a numerical value. Then add 1 to both sides. The equation becomes \(x=(\log_{2} (16/6)+1)/3\).
04

Compute the Final Answer

Evaluate the expression to get the final answer. It is important to round off the answer to three decimal places as indicated in the problem statement.

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Most popular questions from this chapter

Use the Richter scale \(R=\log \frac{l}{I_{0}}\) for measuring the magnitudes of earthquakes. Find the intensity \(I\) of an earthquake measuring \(R\) on the Richter scale (let \(I_{0}=1\) ). (a) Southern Sumatra, Indonesia in \(2007, R=8.5\) (b) Illinois in \(2008, R=5.4\) (c) Costa Rica in \(2009, R=6.1\)

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$-x e^{-x}+e^{-x}=0$$

An exponential growth model has the form ________ and an exponential decay model has the form ________.

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers \(y\) of cell sites from 1985 through 2008 can be modeled by \(y=\frac{237,101}{1+1950 e^{-0.355 t}}\) where \(t\) represents the year, with \(t=5\) corresponding to \(1985 .\) (Source: CTIA-The Wireless Association) (a) Use the model to find the numbers of cell sites in the years 1985,2000 , and 2006 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the number of cell sites will reach 235,000 . (d) Confirm your answer to part (c) algebraically.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{2}(2 x-3)=\log _{2}(x+4)$$
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