Given that log₅ x = a and log₅ y = b, find log₅ (x²/25) in terms of a and b.

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To find log₅ ((x²)/(25)) in terms of a and b, we will use the properties of logarithms. Given: log₅ x = a; log₅ y = b We need to simplify log₅ ((x²)/(25)). Step 1: Apply the quotient rule of logarithms, log_c (M/N) = log_c M − log_c N. log₅ ((x²)/(25)) = log₅ (x²) − log₅ (25); Step 2: Apply the power rule of logarithms, log_c (M^k) = k log_c M, to the first term. log₅ (x²) = 2 log₅ x; Also, express 25 as a power of 5: 25 = 5². log₅ (25) = log₅ (5²) Step 3: Substitute these into the expression from Step 1. log₅ ((x²)/(25)) = 2 log₅ x − log₅ (5²); Step 4: Apply the power rule again to the second term and use the identity log_c c = 1. log₅ (5²) = 2 log₅ 5 = 2 × 1 = 2; Step 5: Substitute the given value log₅ x = a and the result from Step 4 into the expression. log₅ ((x²)/(25)) = 2a − 2 The value b is not used in this particular expression. The simplified expression is 2a − 2.

Given that log₅ x = a and log₅ y = b, find log₅ (x²/25) in terms of a and b.

Answer

Step-by-step solution

Apply the quotient rule of logarithms, $\log_c \left(\frac{M}{N}\right) = \log_c M - \log_c N$ .

Apply the power rule of logarithms, $\log_c (M^k) = k \log_c M$ , to the first term.

Substitute these into the expression from Step 1.

Apply the power rule again to the second term and use the identity $\log_c c = 1$ .

Substitute the given value $\log_5 x = a$ and the result from Step 4 into the expression.

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Given that log₅ x = a and log₅ y = b, find log₅ (x²/25) in terms of a and b.

Answer

Step-by-step solution

Apply the quotient rule of logarithms, log⁡c(MN)=log⁡cM−log⁡cN\log_c \left(\frac{M}{N}\right) = \log_c M - \log_c Nlogc​(NM​)=logc​M−logc​N.

Apply the power rule of logarithms, log⁡c(Mk)=klog⁡cM\log_c (M^k) = k \log_c Mlogc​(Mk)=klogc​M, to the first term.

Substitute these into the expression from Step 1.

Apply the power rule again to the second term and use the identity log⁡cc=1\log_c c = 1logc​c=1.

Substitute the given value log⁡5x=a\log_5 x = alog5​x=a and the result from Step 4 into the expression.

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More Mathematics Questions

Apply the quotient rule of logarithms, $\log_c \left(\frac{M}{N}\right) = \log_c M - \log_c N$ .

Apply the power rule of logarithms, $\log_c (M^k) = k \log_c M$ , to the first term.

Apply the power rule again to the second term and use the identity $\log_c c = 1$ .

Substitute the given value $\log_5 x = a$ and the result from Step 4 into the expression.