We will simplify the expression $3 \log 81 + 2 \log \left(\frac{1}{32}\right)$ step by step.

Step 1: Use the logarithmic power rule

The power rule for logarithms states that $a \log b = \log b^a$. We apply this rule to both terms.

$3 \log 81 = \log 81^3$ $2 \log \left(\frac{1}{32}\right) = \log \left(\frac{1}{32}\right)^2 = \log \left(\frac{1}{1024}\right)$

Step 2: Simplify the logarithmic terms

Now we calculate the values inside the logarithms:

• $81^3 = 531,441$
• $\left(\frac{1}{32}\right)^2 = \frac{1}{1024}$

So the expression becomes: $\log 531,441 + \log \left(\frac{1}{1024}\right)$

Step 3: Use the logarithmic addition rule

The addition rule for logarithms states that $\log a + \log b = \log (a \times b)$. We can now combine the two terms:

$\log \left(531,441 \times \frac{1}{1024}\right)$

Step 4: Multiply the values inside the logarithm

$531,441 \times \frac{1}{1024} = 519.5703125$

Thus, the expression simplifies to: $\log 519.5703125$

This is the simplified form. If you need a decimal approximation, you can calculate $\log 519.5703125$, which is approximately $2.715$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How can the power rule for logarithms help in simplifying logarithmic expressions?
2. What is the relationship between logarithms and exponents?
3. How can you use logarithms to solve equations involving exponential growth?
4. What are the properties of logarithms that simplify complex expressions?
5. Can logarithmic identities be used to evaluate expressions without a calculator?

Tip: Always remember to check the base of logarithms; if none is specified, it's usually base 10 (common logarithm).