We are given the exponential equation 2 to the power of x equals 56. Our goal is to find the value of x, rounded to the nearest hundredth, which means two decimal places. This type of equation cannot be solved by simple inspection because 56 is not a power of 2 that we recognize immediately. Therefore, we need to use logarithms to solve for the exponent.
To solve this exponential equation, we need to apply the inverse operation of exponentiation, which is taking the logarithm. We take the logarithm base 2 of both sides. On the left side, log base 2 of 2 to the power x simplifies to x, because logarithm and exponentiation are inverse operations. This leaves us with x equals log base 2 of 56.
Most calculators don't have a direct button for logarithm base 2. To calculate log base 2 of 56, we use the change of base formula. This formula states that log base a of b equals the natural logarithm of b divided by the natural logarithm of a. So log base 2 of 56 equals ln of 56 divided by ln of 2. Alternatively, we can use common logarithm base 10: log of 56 divided by log of 2. Both methods will give us the same result.
Now we calculate the values. The natural logarithm of 56 is approximately 4.0254, and the natural logarithm of 2 is approximately 0.6931. We then divide these values: 4.0254 divided by 0.6931, which gives us approximately 5.807. Let's verify this using common logarithms. Log base 10 of 56 is approximately 1.7482, and log base 10 of 2 is approximately 0.3010. Dividing these gives 1.7482 divided by 0.3010, which also equals approximately 5.807.
We need to round our answer to the nearest hundredth. Our calculated value is approximately 5.807. Rounding to two decimal places gives us x equals 5.81. To verify this solution, we substitute back into the original equation. We calculate 2 to the power of 5.81, which equals approximately 56.3. This is very close to 56, confirming our solution is correct. The small difference is due to rounding our value of x to two decimal places.
Let's explore an alternative approach using a graphical method. We can plot the function y equals 2 to the power of x and the horizontal line y equals 56. The solution to our equation is the x-coordinate of the point where these two graphs intersect. When we plot these functions, we see that they intersect at approximately x equals 5.81. This confirms our algebraic solution. To summarize the key steps: First, we took the logarithm of both sides of the equation. Second, we applied the change of base formula to convert to natural or common logarithms. Third, we performed the numerical calculation. Finally, we rounded the result to the required precision of two decimal places.