Question $\log _{a} b^{3}+4 \log _{a}\left(a c^{3}\right)-7$ , where $a, b, c>1$ , written as a single logarithm, is $\log _{a}\left(b^{3} c^{3}\right)$ $\log _{a}\left(\frac{b c}{a^{4}}\right)$ $\log _{a}\left(\frac{b^{3} c^{12}}{a^{4}}\right)$ $\log _{a}\left(\frac{b^{3} c^{12}}{a^{3}}\right)$

Studdy Solution

STEP 1

1. We are given a logarithmic expression and need to simplify it into a single logarithm.
2. The properties of logarithms, such as the power rule, product rule, and quotient rule, will be used to simplify the expression.

STEP 2

1. Apply the power rule to simplify the logarithms.
2. Use the product rule to combine logarithms.
3. Simplify the expression by combining terms and using the quotient rule.

STEP 3

Apply the power rule to each logarithmic term. The power rule states that $\log_a(x^n) = n \cdot \log_a(x)$ .
$\log_a(b^3) = 3 \cdot \log_a(b)$
$4 \log_a(ac^3) = 4 \cdot (\log_a(a) + \log_a(c^3)) = 4 \cdot (1 + 3 \cdot \log_a(c)) = 4 + 12 \cdot \log_a(c)$

STEP 4

Combine the logarithms using the product rule. The product rule states that $\log_a(x) + \log_a(y) = \log_a(xy)$ .
Combine the terms:
$3 \cdot \log_a(b) + 12 \cdot \log_a(c) + 4 - 7$
Simplify the constant terms:
$= 3 \cdot \log_a(b) + 12 \cdot \log_a(c) - 3$

STEP 5

Express the simplified expression as a single logarithm using the quotient rule. The quotient rule states that $\log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right)$ .
Combine the logarithmic terms:
$= \log_a(b^3) + \log_a(c^{12}) - \log_a(a^3)$
Use the product and quotient rules:
$= \log_a\left(\frac{b^3 \cdot c^{12}}{a^3}\right)$
The expression written as a single logarithm is:
$\boxed{\log_a\left(\frac{b^3 c^{12}}{a^3}\right)}$

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