If f(x) = `log_(10) (1 + x)` than what is 4f (4) + 5f(1) - `log_(10)2` equal to ?

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem step by step, we start with the function given:

**Step 1: Define the function and find f(4) and f(1)**

The function is defined as:
\[ f(x) = \log_{10}(1 + x) \]

Now, we need to find \( f(4) \) and \( f(1) \).

- For \( f(4) \):
\[ f(4) = \log_{10}(1 + 4) = \log_{10}(5) \]

- For \( f(1) \):
\[ f(1) = \log_{10}(1 + 1) = \log_{10}(2) \]

**Step 2: Substitute \( f(4) \) and \( f(1) \) into the expression**

We need to evaluate:
\[ 4f(4) + 5f(1) - \log_{10}(2) \]

Substituting the values we found:
\[ 4f(4) = 4 \cdot \log_{10}(5) = \log_{10}(5^4) = \log_{10}(625) \]
\[ 5f(1) = 5 \cdot \log_{10}(2) = \log_{10}(2^5) = \log_{10}(32) \]

Now, substituting these into the expression:
\[ \log_{10}(625) + \log_{10}(32) - \log_{10}(2) \]

**Step 3: Use logarithm properties to combine the logs**

Using the property of logarithms that states \( \log_a(b) + \log_a(c) = \log_a(b \cdot c) \):
\[ \log_{10}(625) + \log_{10}(32) = \log_{10}(625 \cdot 32) \]

Now, we can simplify:
\[ 625 \cdot 32 = 20000 \]

So, we have:
\[ \log_{10}(20000) - \log_{10}(2) \]

Using the property \( \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \):
\[ \log_{10}\left(\frac{20000}{2}\right) = \log_{10}(10000) \]

**Step 4: Simplify further**

Since \( 10000 = 10^4 \):
\[ \log_{10}(10000) = \log_{10}(10^4) = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]