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Problem 752

Find log $$_{3} 729 .$$

Expert verified
The short answer is: $$\log_{3}{729} = 6$$
See the step by step solution

Step 1: Identify base and desired result

The base in this logarithmic expression is 3, and we want to find the exponent (x) to which it must be raised to obtain 729.

Step 2: Write the logarithmic equation

Express the problem as a logarithmic equation: $$\log_{3}{729} = x$$.

Step 3: Convert to exponential form

Rewrite the logarithmic equation into its equivalent exponential form: $$3^x = 729$$.

Step 4: Solve for x

In this step, we need to find the value of x that satisfies the equation $$3^x = 729$$. We can write 729 as a power of 3: $$729 = 3^6$$ (since $$3 * 3 * 3 * 3 * 3 * 3 = 729$$). Now we can rewrite the equation as: $$3^x = 3^6$$. Since the bases are the same, we can set the exponents equal to each other: $$x = 6$$.

Step 5: State the final answer

Now we know that $$\log_{3}{729} = 6$$.

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