Math  /  Algebra

Questionlogab3+4loga(ac3)7\log _{a} b^{3}+4 \log _{a}\left(a c^{3}\right)-7, where a,b,c>1a, b, c>1, written as a single logarithm, is loga(b3c3)\log _{a}\left(b^{3} c^{3}\right) loga(bca4)\log _{a}\left(\frac{b c}{a^{4}}\right) loga(b3c12a4)\log _{a}\left(\frac{b^{3} c^{12}}{a^{4}}\right) loga(b3c12a3)\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 loga(xn)=nloga(x)\log_a(x^n) = n \cdot \log_a(x).
loga(b3)=3loga(b)\log_a(b^3) = 3 \cdot \log_a(b)
4loga(ac3)=4(loga(a)+loga(c3))=4(1+3loga(c))=4+12loga(c)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 loga(x)+loga(y)=loga(xy)\log_a(x) + \log_a(y) = \log_a(xy).
Combine the terms:
3loga(b)+12loga(c)+473 \cdot \log_a(b) + 12 \cdot \log_a(c) + 4 - 7
Simplify the constant terms:
=3loga(b)+12loga(c)3= 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 loga(x)loga(y)=loga(xy)\log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right).
Combine the logarithmic terms:
=loga(b3)+loga(c12)loga(a3)= \log_a(b^3) + \log_a(c^{12}) - \log_a(a^3)
Use the product and quotient rules:
=loga(b3c12a3)= \log_a\left(\frac{b^3 \cdot c^{12}}{a^3}\right)
The expression written as a single logarithm is:
loga(b3c12a3)\boxed{\log_a\left(\frac{b^3 c^{12}}{a^3}\right)}

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