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Chapter 3: Problem 121

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Short Answer

Expert verified
To solve an exponential equation when both sides can be written as a power of the same base, rewrite each side of the equation as powers of the same base, set the exponents equal to each other, and then solve the resulting equation for the unknown variable.

Step by step solution

01

Rewrite the equation

The first step in solving an exponential equation is rewriting both sides of the equation as powers of the same base. If you can write each side of the equation with the same base, then you can set the exponents equal to each other.
02

Set the exponents equal to each other

Once both sides of the equation are written as a power of the same base, you can set the exponents equal to each other. Because if \(a^m = a^n\), then \(m = n\). This gives you a new equation that you can solve.
03

Solve for the unknown

Lastly, solve the new equation for the variable. This could involve isolating the variable on one side of the equation or using algebraic methods such as factoring, expanding, or using the quadratic formula if necessary.

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Most popular questions from this chapter

Consider the quadratic function $$ f(x)=-4 x^{2}-16 x+3 $$ a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

Solve each equation in Exercises \(146-148 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

Use a calculator to evaluate \(\left(1+\frac{1}{x}\right)^{x}\) for \(x=10,100,1000\) \(10,000,100,000,\) and \(1,000,000 .\) Describe what happens to the expression as \(x\) increases.

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$

One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.
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