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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 33
Chapter 5, Problem 33

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e(1−5x)=793

Verified step by step guidance
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Start with the given exponential equation: \(e^{(1 - 5x)} = 793\).
To solve for \(x\), take the natural logarithm (ln) of both sides to utilize the property that \(\ln(e^y) = y\). This gives: \(\ln\left(e^{(1 - 5x)}\right) = \ln(793)\).
Simplify the left side using the logarithm property: \(1 - 5x = \ln(793)\).
Isolate the term containing \(x\) by subtracting 1 from both sides: \(-5x = \ln(793) - 1\).
Finally, solve for \(x\) by dividing both sides by \(-5\): \(x = \frac{1 - \ln(793)}{5}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Equations

An exponential equation involves variables in the exponent, such as e^(1−5x) = 793. Solving these requires isolating the exponential expression and then applying logarithms to both sides to solve for the variable.
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Solving Exponential Equations Using Logs

Natural Logarithms

The natural logarithm (ln) is the inverse function of the exponential function with base e. Applying ln to both sides of an equation like e^(1−5x) = 793 allows you to simplify the exponent and solve for x.
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The Natural Log

Using a Calculator for Approximations

After expressing the solution in logarithmic form, a calculator is used to find decimal approximations. This step involves evaluating logarithms and performing arithmetic to get a numerical answer rounded to the desired decimal places.
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Solving Exponential Equations Using Logs
Related Practice
Textbook Question

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2.2x

Textbook Question

In Exercises 32–35, the graph of a logarithmic function is given. Select the function for each graph from the following options: f(x) = log x, g(x) = log(-x), h(x) = log(2-x), r(x)= 1+log(2-x)

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Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e(5x−3) - 2 =10,476

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Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb(xy3z3)\(\log\)_{b}\(\left\)(\(\frac{\sqrt{x}\)y^3}{z^3}\(\right\))

Textbook Question

Evaluate each expression without using a calculator. log5 5

Textbook Question

Evaluate each expression without using a calculator. log64 8