Problem 14
Question
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{2-x}=\frac{1}{125}$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x = 5\).
1Step 1: Express the equation with the same base
We see that we have base 5 on the left side, and \(\frac{1}{125}\) on the right side. \(\frac{1}{125}\) can be expressed as \(5^{-3}\) because \(5^{3} = 125\), and therefore \(\frac{1}{125} = 5^{-3}\). So, the equation is rewritten as: 5^{2-x} = 5^{-3}
2Step 2: Equate the powers and solve for x
Since the bases on both sides of the equation are equal, the equation becomes: 2-x = -3. Solving for x gives: x = 2 - (-3) = 5
Other exercises in this chapter
Problem 13
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calcula
View solution Problem 13
Write each equation in its equivalent logarithmic form. $$\sqrt[3]{8}=2$$
View solution Problem 14
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calcula
View solution Problem 15
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$6^{\frac{x-3}{4}}=\sqrt{6}$$
View solution