Evaluate log(10^4⋅8)

Question

.(1 point)
Responses

log(10^4⋅8)=log(4+8)
logarithm left parenthesis 10 superscript 4 baseline times 8 right parenthesis equals logarithm left parenthesis 4 plus 8 right parenthesis

log(10^4⋅8)=4+log8
logarithm left parenthesis 10 superscript 4 baseline times 8 right parenthesis equals 4 plus logarithm 8

log(10^4⋅8)=log(4⋅8)
logarithm left parenthesis 10 superscript 4 baseline times 8 right parenthesis equals logarithm left parenthesis 4 times 8 right parenthesis

log(10^4⋅8)=log4+8

Answer 1

To evaluate \( \log(10^4 \cdot 8) \), we can use the properties of logarithms.

1. Product Rule: The logarithm of a product is the sum of the logarithms:
   \[ \log(a \cdot b) = \log a + \log b \]

Using this property, we can rewrite \( \log(10^4 \cdot 8) \) as follows: \[ \log(10^4 \cdot 8) = \log(10^4) + \log(8) \]

2. Power Rule: The logarithm of a power is the exponent times the logarithm of the base:
   \[ \log(a^b) = b \cdot \log(a) \]

Thus, we can simplify \( \log(10^4) \): \[ \log(10^4) = 4 \cdot \log(10) = 4 \]

3. Putting it all together: Now substitute back: \[ \log(10^4 \cdot 8) = \log(10^4) + \log(8) = 4 + \log(8) \]

So, the final evaluation is: \[ \log(10^4 \cdot 8) = 4 + \log(8) \]

From the responses provided, the correct one is: \[ \log(10^4 \cdot 8) = 4 + \log 8 \]