We are given the exponential equation 2 to the power of x equals 32. Our goal is to solve for the unknown exponent x and express the answer to the nearest hundredth.
Let's solve this by recognizing powers of 2. We know that 2 to the first power is 2, 2 squared is 4, 2 cubed is 8, 2 to the fourth is 16, and 2 to the fifth is 32. So we can rewrite our equation as 2 to the x equals 2 to the fifth. Since the bases are equal, the exponents must be equal, giving us x equals 5. To the nearest hundredth, this is 5.00.
Now let's solve this using logarithms, which works for any exponential equation. Starting with 2 to the x equals 32, we take the natural logarithm of both sides. Using the logarithm property that ln of a to the b equals b times ln of a, we get x times ln of 2 equals ln of 32. Solving for x gives us x equals ln of 32 divided by ln of 2. Calculating numerically, ln of 32 is approximately 3.4657, and ln of 2 is approximately 0.6931. Dividing these gives x approximately 5.000, which confirms our previous answer of x equals 5.00.
Let's verify our solution by substituting x equals 5 back into the original equation. 2 to the 5th power is indeed 32, confirming our answer. For rounding to the nearest hundredth, since our exact answer is 5, we express it as 5.00. On a number line, 5.000 is exactly at the hundredth mark, so no rounding is needed. Our final answer is x equals 5.00.
Let's summarize our two solution methods. Method 1 involves recognizing powers: express the number as a power of the base, set the exponents equal, and solve. Method 2 uses logarithms: take the log of both sides, apply the logarithm property, and solve algebraically. Use Method 1 when the number is a recognizable power of the base, and Method 2 for any exponential equation. Both methods lead us to the same final answer: x equals 5.00.