Question

(Need asap)Use the laws of logarithms to express each of the following as a single logarithm.
a. log5 2+log2 3
b. log 12-log 3
c. ln 4+2 ln 3
d. log M+2 log N
e. logb M+logb N+logb P

Answer 1

Answer:

a)$\log_5(2) + \log_5(3) = \log_5(6)$

b) $\log(12) - \log(3) = \log(4)$

c) $\ln(4) + 2\ln(3) = \ln(36)$

d) $\sf \log(M) + 2\log(N) = \log(MN^2)$

e)$\sf \log_b(M) + \log_b(N) + \log_b(P) = \log_b(M \cdot N \cdot P)$

Step-by-step explanation:

The rules of logarithms are:

Product Rule: $\log_b(a) + \log_b(c) = \log_b(ac)$

Quotient Rule: $\log_b(a) - \log_b(c) = \log_b\left(dfrac{a}{c}\right)$

Power Rule: $n \cdot \log_b(a) = \log_b(a^n)$

Now, let's apply these rules to each problem:

a. $\log_5(2) + \log_5(3)$

Using the product rule, we have:

$\log_5(2) + \log_5(3) = \log_5(2 \cdot 3) \\= \log_5(6)$

$\dotfill$

b. $\log(12) - \log(3)$

Using the quotient rule, we have:

$\log(12) - \log(3) = \log\left(\dfrac{12}{3}\right)\\ = \log(4)$

$\dotfill$

c. $\ln(4) + 2\ln(3)$

Using the power rule, we have:

$\ln(4) + 2\ln(3) = \ln(4) + \ln(3^2) \\= \ln(4 \cdot 3^2) = \ln(36)$

$\dotfill$

d. $\sf \log(M) + 2\log(N)$

Using the power rule, we have:

$\sf \log(M) + 2\log(N) = \log(M) + \log(N^2)\\\sf = \log(MN^2)$

$\dotfill$

e. $\sf \log_b(M) + \log_b(N) + \log_b(P)$

Using the product rule multiple times, we have:

$\sf \log_b(M) + \log_b(N) + \log_b(P) = \log_b(M\cdot N) + \log_b(P)\\\sf = \log_b(M \cdot N \cdot P)$