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Chapter 3: Problem 58

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$8\left(4^{6-2 x}\right)+13=41$$

Short Answer

Expert verified
The solution to the exponential equation, approximated to three decimal places, is: \[x = 4.433\].

Step by step solution

01

Isolate the Exponential Term

Subtract 13 from both sides of the equation, we get: \[8\left(4^{6-2x}\right) = 41 - 13 = 28\] \[8\left(4^{6-2x}\right) = 28\]. Now we can divide both sides by 8 to isolate the exponential term \[4^{6-2x} = \frac{28}{8} = 3.5\].
02

Convert the Exponential Equation to a Logarithmic Equation

To rewrite the equation \[4^{6 - 2x} = 3.5\] in logarithmic form, we can use the relationship: \[b^y = x \leftrightarrow \log_b x = y\], where 'b' is the base, 'x' is the result and 'y' is the exponent. Therefore, the equation can be rewritten as: \[\log_4 3.5 = 6 - 2x\].
03

Solve for 'x'

To solve for 'x', first, isolate 'x to one side of the equation which would result in: \[-2x = \log_4 3.5 - 6\] Then divide both sides of the equation by -2 to get 'x': \[x = \frac{\log_4 3.5 - 6}{-2}\]
04

Approximate 'x' to Three Decimal Places

Calculate the value of 'x' using a calculator or logarithmic table. Remember that the answer should be approximated to three decimal places.

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Most popular questions from this chapter

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.

After discontinuing all advertising for a tool kit in \(2004,\) the manufacturer noted that sales began to drop according to the model \(S=\frac{500,000}{1+0.4 e^{k t}}\) where \(S\) represents the number of units sold and \(t=4\) represents \(2004 .\) In \(2008,\) the company sold 300,000 units. (a) Complete the model by solving for \(k\). (b) Estimate sales in 2012 .

Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$3-\ln x=0$$

Write two or three sentences stating the general guidelines that you follow when solving (a) exponential equations and (b) logarithmic equations.
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