`log_(10)a^p.b^q.c^r=?`

Options

A. a) `pqr(log_(10)abc)`
B. b) `plog_(10)a+qlog_(10)b+rlog_(10)c`
C. c) `pqr(log_(10)a+log_(10)b+log_(10)c)`
D. d) `(log_(10)a)^p+(log_(10)b)^q+(log_(10)c)^r`

Answer

To solve the expression \( \log_{10}(a^p \cdot b^q \cdot c^r) \), we can use the properties of logarithms. Here’s a step-by-step breakdown:

### Step 1: Apply the Product Rule of Logarithms
The product rule states that the logarithm of a product is the sum of the logarithms. Therefore, we can express the logarithm of the product \( a^p \cdot b^q \cdot c^r \) as:

\[ \log_{10}(a^p \cdot b^q \cdot c^r) = \log_{10}(a^p) + \log_{10}(b^q) + \log_{10}(c^r) \]

**Hint:** Remember that the logarithm of a product can be split into the sum of the logarithms of the individual factors.

### Step 2: Apply the Power Rule of Logarithms
The power rule states that the logarithm of a number raised to a power is the power times the logarithm of the number. Applying this rule to each term, we get:

\[ \log_{10}(a^p) = p \cdot \log_{10}(a) \]
\[ \log_{10}(b^q) = q \cdot \log_{10}(b) \]
\[ \log_{10}(c^r) = r \cdot \log_{10}(c) \]

### Step 3: Combine the Results
Now, substituting these results back into our expression from Step 1, we have:

\[ \log_{10}(a^p \cdot b^q \cdot c^r) = p \cdot \log_{10}(a) + q \cdot \log_{10}(b) + r \cdot \log_{10}(c) \]

### Final Answer
Thus, the final expression for \( \log_{10}(a^p \cdot b^q \cdot c^r) \) is:

\[ \log_{10}(a^p \cdot b^q \cdot c^r) = p \cdot \log_{10}(a) + q \cdot \log_{10}(b) + r \cdot \log_{10}(c) \]

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