Problem 94
Question
Solve each equation. $$ 3^{x+2} \cdot 3^{x}=81 $$
Step-by-Step Solution
Verified Answer
The solution to the equation \(3^{x+2} \cdot 3^{x}=81\) is \(x=1\).
1Step 1: Combine the exponential expressions
The given equation is \(3^{x+2} \cdot 3^{x}=81\). The base in both expressions is the same (3), which means the powers can be added together. Therefore the first step is to combine the powers to get \(3^{2x+2} = 81\).
2Step 2: Change the similar bases
The right side of the equation, 81, can be expressed as a power of 3. Therefore, rewrite 81 as \(3^4\). This results in the equation \(3^{2x+2} = 3^4\).
3Step 3: Equate the Powers
Because the bases are the same, the powers must also be equal (assuming the base is non-zero). That gives the equation \(2x + 2 = 4\).
4Step 4: Solve for the variable
Now you have simple linear equation. Subtract 2 from both sides to get \(2x = 2\), then divide by 2 to solve for \(x\), to get \(x=1\).
Other exercises in this chapter
Problem 93
In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 125} $$
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The hyperbolic cosine and hyperbolic sine functions are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \quad \text { and } \quad \sinh x=\frac{e^{x}-e^{-x}}{2} $$
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Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s)
View solution Problem 94
In Exercises 81–100, evaluate or simplify each expression without using a calculator. $$ e^{\ln 300} $$
View solution