Write as one logarithm: (a) 3 loga u-loga v; (b) (1)/(3) loga 5-3 loga x-4 loga y; (c) (1)/(3) l

step 1 We are going to do problem number 9. In this question we have to write the logarithm in one expr

Write as one logarithm: (a) 3 loga u-loga v; (b) (1)/(3)...

Write as one logarithm: (a) $3 \log _a u-\log _a v$; (b) $\frac{1}{3} \log _a 5-3 \log _a x-4 \log _a y$; (c) $\frac{1}{3} \log _a(x-3)+3 \log _a x+2 \log _a(1+x)$; (d) $\frac{1}{2}\left[\log _a x+3 \log _a y-5 \log _a(z-2)\right]$; (e) $\frac{1}{2} \ln (x+1)-\frac{1}{2} \ln (x-1)+\ln C$

Write as one logarithm:
(a) $3 \log _a u-\log _a v$;
(b) $\frac{1}{3} \log _a 5-3 \log _a x-4 \log _a y$;
(c) $\frac{1}{3} \log _a(x-3)+3 \log _a x+2 \log _a(1+x)$;
(d) $\frac{1}{2}\left[\log _a x+3 \log _a y-5 \log _a(z-2)\right]$;
(e) $\frac{1}{2} \ln (x+1)-\frac{1}{2} \ln (x-1)+\ln C$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Power Rule of Logarithms

The power rule states that multiplying a logarithm by a constant is equivalent to raising its argument to the power of that constant, i.e., k·log_b(M) is the same as log_b(M^k). This is useful for rewriting coefficients as exponents inside the logarithm, which facilitates the combination of terms when simplifying expressions.

Product Rule of Logarithms

The product rule tells us that the sum of two logarithms with the same base can be combined into a single logarithm whose argument is the product of the original arguments, expressed as log_b(M) + log_b(N) = log_b(M·N). This property is central when combining logarithmic terms that are added together.

Quotient Rule of Logarithms

The quotient rule indicates that the difference of two logarithms with the same base can be rewritten as a single logarithm with the argument being the quotient of the original arguments, i.e., log_b(M) - log_b(N) = log_b(M/N). This rule is essential for merging logarithms that are subtracted from each other into one concise expression.

Transcript

Please give Ace some feedback