Answer :
Answer:
D. log₁₀ 1,000.
E. log₃ 27
Step-by-step explanation:
Logarithms
Logarithms are the inverse equations of their exponential counterparts.
In an exponential equation, it finds the value of an exponent made a base and power.
In a logarithm equation, it finds the value of the power that produces an exponent with a certain base.
In other words,
[tex]b^x=c[/tex]
[tex]log_b(c)=x[/tex].
Knowing that logs are inverses of exponents can help evaluate them.
[tex]\hrulefill[/tex]
Solving the Problem
To determine the correct answers, we must find the value of [tex]log_2=8[/tex].
This logarithm means that when 2 is raised to some power x, it produces 8.
[tex]2^x=8[/tex]
Knowing 8 to be the cube value of 2,
[tex]x=3 \rightarrow log_2(8)=3[/tex].
Now, we have to find answer choices where their value is also 3.
[tex]\dotfill[/tex]
Narrowing Down the Answer Choices
Using the facts mentioned in the top section, we can rewrite all of the logarithms as exponents.
[tex]A)\: 5^x=20\\\\B)\: 10^x=125\\\\C)\: 10^x=100\\\\[/tex]
[tex]D)\: 10^x=1000\\\\E)\: 3^x=27\\\\F)\: 10^x=0.001\\[/tex]
No power that raises 5 will get 20, so we can eliminate A.
The same can be said for B.
100 is the squared value of 10, meaning that x = 2. This isn't the same as the solution x = 3, so C is crossed off.
1000 is the cubed value of 10, meaning x = 3. So, D is a correct answer.
Same thing with E, 27 is the cubed value of 3, so E is also correct.
0.001 or [tex]\dfrac{1}{1000}[/tex] is the same as [tex]10^-^3[/tex], meaning x = -3. Not 3, so F is wrong.
Thus, D and E are our final answers.
