Problem 83 - College Algebra Essentials

Question

Evaluate or simplify each expression without using a calculator. $$\log 10^{7}$$

Step-by-Step Solution

Answer

The expression \(\log_{10} 10^{7}\) simplifies to 7.

Step 1: Apply Logarithmic Property

Remember the property of logarithms that states \(\log_b(b^x) = x\). Applying the property, the expression \(\log_{10} 10^{7}\) simplifies directly to the exponent value as the base 10 and the number is 10 raised to the power of 7. Therefore, \(\log_{10} 10^{7}\) simplifies to 7.

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The expression \(\log_{10} 10^{7}\) simplifies to 7.