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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 35
Chapter 5, Problem 35

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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Start with the given equation: \(e^{(5x - 3)} - 2 = 10476\).
Isolate the exponential term by adding 2 to both sides: \(e^{(5x - 3)} = 10476 + 2\).
Simplify the right side: \(e^{(5x - 3)} = 10478\).
Take the natural logarithm (ln) of both sides to undo the exponential: \(\ln\left(e^{(5x - 3)}\right) = \ln(10478)\).
Use the logarithm power rule to bring down the exponent: \((5x - 3) = \ln(10478)\), then solve for \(x\) by isolating it: \(5x = \ln(10478) + 3\), and finally \(x = \frac{\ln(10478) + 3}{5}\).

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Key Concepts

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

Exponential Equations

Exponential equations involve variables in the exponent, such as e^(5x−3). 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 and Common Logarithms

Natural logarithms (ln) use base e, while common logarithms (log) use base 10. They are inverse functions of exponential functions and are used to solve equations where the variable is an exponent.
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Graphs of Common Functions

Using a Calculator for Approximation

After expressing the solution in logarithmic form, calculators help find decimal approximations. This step involves evaluating logarithms and rounding the result to the desired decimal places, such as two decimals.
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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

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5x2y243\(\log\)_5 \(\sqrt\)[3]{\(\frac{x^2 y}{24}\)}

Textbook Question

Evaluate each expression without using a calculator. log5 5

Textbook Question

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = ex-1

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(1−5x)=793