Simplify/Condense log base 5 of 1/250

Problem

(log_5)(1/250)

Solution

1. Rewrite the fraction using a negative exponent to simplify the logarithmic argument.

(log_5)(1/250)=(log_5)(250(−1))

2. Apply the power rule for logarithms, (log_b)(xn)=n*(log_b)(x) to move the exponent to the front.

−1⋅(log_5)(250)

3. Factor the number 250 into a product involving powers of the base 5

250=125⋅2=5⋅2

4. Substitute the factored form back into the expression.

−(log_5)(5⋅2)

5. Apply the product rule for logarithms, (log_b)(x*y)=(log_b)(x)+(log_b)(y)

−((log_5)(5)+(log_5)(2))

6. Evaluate the term (log_5)(5) using the property (log_b)(bx)=x

−(3+(log_5)(2))

7. Distribute the negative sign to reach the final condensed form.

−3−(log_5)(2)

Final Answer

(log_5)(1/250)=−3−(log_5)(2)

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