Math  /  Algebra

QuestionSolve the logarithmic equation. Don't forget to check your solution. [*Don't change a fraction to a decimal applicable.] log5(x)=1\log _{5}(x)=1
Exact solution: x=x=

Studdy Solution

STEP 1

What is this asking? This problem is asking us to find the value of xx that makes the equation log5(x)=1\log_5(x) = 1 true.
It's like asking, "5 raised to what power equals xx?". Watch out! Remember, the logarithm of a negative number or zero is undefined, so watch out for those sneaky solutions!

STEP 2

1. Rewrite the logarithmic equation in exponential form.
2. Calculate the result.

STEP 3

Logarithms and exponents are **inverse operations**, like multiplication and division.
The equation logb(x)=y\log_b(x) = y means the same thing as by=xb^y = x.
The **base** of the logarithm becomes the **base** of the exponent, the result of the logarithm becomes what the exponential expression is **equal to**, and the input of the logarithm becomes the **exponent**.

STEP 4

In our equation, log5(x)=1\log_5(x) = 1, the **base** is 55, the **result** of the logarithm is 11, and the **input** is xx.
So, rewriting this in exponential form gives us 51=x5^1 = x.

STEP 5

We have 51=x5^1 = x.
Any number raised to the power of 11 is just itself.
So, 51=55^1 = 5.

STEP 6

Therefore, x=5x = \mathbf{5}.
This means that log5(5)=1\log_5(5) = 1, which is true because 55 raised to the power of 11 equals 55.

STEP 7

Exact solution: x=5x = 5

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