Respuesta :
Answer:
[tex]\log _3\left(81\right)=4[/tex]
Step-by-step explanation:
Given : [tex]\log _3\left(81\right)[/tex]
We have to find the value of [tex]\log _3\left(81\right)[/tex]
Consider [tex]\log _3\left(81\right)[/tex]
Rewrite 81 in base power form, [tex]81=3^4[/tex]
[tex]=\log _3\left(3^4\right)[/tex]
[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)[/tex]
We have,
[tex]\log _3\left(3^4\right)=4\log _3\left(3\right)[/tex]
[tex]=4\log _3\left(3\right)[/tex]
[tex]\mathrm{Apply\:log\:rule}:\quad \log _a\left(a\right)=1[/tex]
[tex]\log _3\left(3\right)=1[/tex]
[tex]=4\log _3\left(3\right)=4[/tex]
Thus, [tex]\log _3\left(81\right)=4[/tex]
4
Further explanation
Now, we will solve the problem of the logarithm.
Some logarithmic properties used this time are as follows:
[tex]\boxed{ \ \log_{a} a = 1 \ }[/tex]
[tex]\boxed{ \ \log_{a}b^c = c \log_a b \ }[/tex]
Let us calculate the value of [tex]\boxed{ \ \log_3 81 \ }[/tex]
Recall that [tex]\boxed{ \ 81 = 3^4. \ }[/tex]
[tex]\boxed{ \ \log_3 81 = \log_3 3^4 \ }[/tex]
[tex]\boxed{ \ \log_3 81 = 4 \log_3 3 \ }[/tex]
[tex]\boxed{ \ \log_3 81 = 4 \times 1 \ }[/tex]
Thus, the result is [tex]\boxed{\boxed{ \ \log_3 81 = 4 \ }}[/tex]
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Alternative problem:
What is the value of [tex]\boxed{ \ \log_{81} 3 \ ? \ }[/tex]
Recall that [tex]\boxed{ \ 3 \rightarrow (3^4)^{\frac{1}{4}} \rightarrow 81^{\frac{1}{4}}. \ }[/tex]
[tex]\boxed{ \ \log_{81} 3 = \log_{81} 81^{\frac{1}{4}} \ }[/tex]
[tex]\boxed{ \ \log_{81} 3 = \frac{1}{4} \log_{81} 81 \ }[/tex]
[tex]\boxed{ \ \log_{81} 3 = \frac{1}{4} \times 1 \ }[/tex]
Thus, the result is [tex]\boxed{\boxed{ \ \log_{81} 3 = \frac{1}{4} \ }}[/tex]
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Notes:
The relationship between logarithms and exponents is as follows:
[tex]\boxed{ \ \log_{a}b = c \ } \leftrightarrow \boxed{ \ a^c = b \ }[/tex]
- a = base of logarithm
- b = numerous
- c = the result of logarithm
- [tex]\boxed{ \ a > 0, \ b > 0, \ a \neq 1 \ }[/tex]
- logarithmic functions and exponential functions are inverses each other.
Learn more
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Keywords: what is the value of log3 81, logarithmic properties, base, numerous, exponential functions, inverses each other
