Question

$\log_{0.01} 1000 + \log_{0.1} 0.0001$ is equal to :

• A. -2
• B. 3
• C. -5/2
• D. 5/2

Answer 1

Answer: (D)

5/2

Answer 2

To evaluate the expression $\log_{0.01} 1000 + \log_{0.1} 0.0001$, we will evaluate each term separately using the definition and properties of logarithms.

Step 1: Evaluate the first term, $\log_{0.01} 1000$.

We can express the base and the argument as powers of 10: $0.01 = \frac{1}{100} = 10^{-2}$ $1000 = 10^3$

So, the term becomes $\log_{10^{-2}} 10^3$. Using the logarithm property $\log_{b^m} a^n = \frac{n}{m} \log_b a$:

$\log_{10^{-2}} 10^3 = \frac{3}{-2} \log_{10} 10$

Since $\log_{10} 10 = 1$:

$\log_{10^{-2}} 10^3 = -\frac{3}{2} \times 1 = -\frac{3}{2}$

Step 2: Evaluate the second term, $\log_{0.1} 0.0001$.

We can express the base and the argument as powers of 10: $0.1 = \frac{1}{10} = 10^{-1}$ $0.0001 = \frac{1}{10000} = 10^{-4}$

So, the term becomes $\log_{10^{-1}} 10^{-4}$. Using the logarithm property $\log_{b^m} a^n = \frac{n}{m} \log_b a$:

$\log_{10^{-1}} 10^{-4} = \frac{-4}{-1} \log_{10} 10$

Since $\log_{10} 10 = 1$:

$\log_{10^{-1}} 10^{-4} = 4 \times 1 = 4$

Step 3: Add the results from Step 1 and Step 2.

$\log_{0.01} 1000 + \log_{0.1} 0.0001 = -\frac{3}{2} + 4$

To add these values, find a common denominator:

$-\frac{3}{2} + \frac{8}{2} = \frac{-3 + 8}{2} = \frac{5}{2}$

Thus, the value of the expression is $\frac{5}{2}$.