Question

What is the value of log 3 81 ?

Answer 1

Answer:

$\log _3\left(81\right)=4$

Step-by-step explanation:

Given : $\log _3\left(81\right)$

We have to find the value of $\log _3\left(81\right)$

Consider $\log _3\left(81\right)$

Rewrite 81 in base power form, $81=3^4$

$=\log _3\left(3^4\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)$

We have,

$\log _3\left(3^4\right)=4\log _3\left(3\right)$

$=4\log _3\left(3\right)$

$\mathrm{Apply\:log\:rule}:\quad \log _a\left(a\right)=1$

$\log _3\left(3\right)=1$

$=4\log _3\left(3\right)=4$

Thus,  $\log _3\left(81\right)=4$

Answer 2

4

Further explanation

Now, we will solve the problem of the logarithm.

Some logarithmic properties used this time are as follows:

$\boxed{ \ \log_{a} a = 1 \ }$

$\boxed{ \ \log_{a}b^c = c \log_a b \ }$

Let us calculate the value of $\boxed{ \ \log_3 81 \ }$

Recall that $\boxed{ \ 81 = 3^4. \ }$

$\boxed{ \ \log_3 81 = \log_3 3^4 \ }$

$\boxed{ \ \log_3 81 = 4 \log_3 3 \ }$

$\boxed{ \ \log_3 81 = 4 \times 1 \ }$

Thus, the result is $\boxed{\boxed{ \ \log_3 81 = 4 \ }}$

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Alternative problem:

What is the value of $\boxed{ \ \log_{81} 3 \ ? \ }$

Recall that $\boxed{ \ 3 \rightarrow (3^4)^{\frac{1}{4}} \rightarrow 81^{\frac{1}{4}}. \ }$

$\boxed{ \ \log_{81} 3 = \log_{81} 81^{\frac{1}{4}} \ }$

$\boxed{ \ \log_{81} 3 = \frac{1}{4} \log_{81} 81 \ }$

$\boxed{ \ \log_{81} 3 = \frac{1}{4} \times 1 \ }$

Thus, the result is $\boxed{\boxed{ \ \log_{81} 3 = \frac{1}{4} \ }}$

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Notes:

The relationship between logarithms and exponents is as follows:

$\boxed{ \ \log_{a}b = c \ } \leftrightarrow \boxed{ \ a^c = b \ }$

• a = base of logarithm
• b = numerous
• c = the result of logarithm
• $\boxed{ \ a > 0, \ b > 0, \ a \neq 1 \ }$
• logarithmic functions and exponential functions are inverses each other.

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Keywords: what is the value of log3 81, logarithmic properties, base, numerous, exponential functions, inverses each other