We are given the exponential equation 2 to the power of x equals 48. Our goal is to find the value of x accurate to the nearest hundredth, which means two decimal places. Since 48 is not a simple power of 2, we cannot solve this equation by inspection. This is where logarithms become essential, as they allow us to solve for variables in the exponent.
To solve exponential equations, we use the fundamental property of logarithms. If a to the power of x equals b, then x equals the logarithm base a of b. Applying this to our equation, 2 to the power of x equals 48 means x equals the logarithm base 2 of 48. To make this calculable, we take the logarithm of both sides. This gives us log of 2 to the x equals log of 48. Using the logarithm power rule, the exponent x can be brought down as a coefficient, resulting in x times log of 2 equals log of 48. Finally, solving for x gives us x equals log of 48 divided by log of 2.
Most calculators don't have a direct button for logarithms with arbitrary bases like base 2. This is where the change of base formula comes in handy. The change of base formula states that the logarithm base a of b equals the logarithm of b divided by the logarithm of a, where the logarithms in the numerator and denominator can be in any base, typically base 10 or natural logarithm base e. Applying this to our problem, the logarithm base 2 of 48 equals the logarithm of 48 divided by the logarithm of 2. This transforms our expression for x into a form we can calculate directly using a standard calculator.
Now we perform the actual calculations. First, we find the logarithm of 48, which is approximately 1.6812. Next, we find the logarithm of 2, which is approximately 0.3010. We then divide these values: 1.6812 divided by 0.3010. Performing this division gives us approximately 5.5849. It's crucial to maintain as much precision as possible throughout these calculations to ensure our final answer is accurate when we round to the nearest hundredth.
To round 5.5849 to the nearest hundredth, we look at the thousandths digit, which is 4. Since 4 is less than 5, we round down, giving us 5.58. To verify our answer, we calculate 2 to the power of 5.58. This gives us approximately 47.86, which is very close to our target value of 48. This confirms that our solution is correct. Therefore, the final answer is x is approximately 5.58.
For additional verification, let's check values immediately adjacent to our answer. Calculating 2 to the power of 5.57 gives approximately 47.55. Our calculated value 2 to the power of 5.58 gives approximately 47.86. And 2 to the power of 5.59 gives approximately 48.17. Comparing these results, we see that 47.86 is indeed the closest to our target of 48. This further confirms that x equals 5.58 is the correct answer to the nearest hundredth.