Condense logarithmic expressions

Condensing logarithmic expressions means reversing the product, quotient, and power properties to rewrite sums, differences, and coefficients of logarithms with the same base as one logarithm, such as $\log_b M+\log_b N=\log_b(MN)$ and $k\log_b M=\log_b(M^k)$ . The reasoning requires preserving equivalent values while ensuring every logarithm’s argument is positive and avoiding the misconception that a sum of logarithms equals the logarithm of a sum; this structure supports solving logarithmic equations.

Detailed Explanation: Condense logarithmic expressions

To condense logarithms, reverse the logarithm properties:

$k\log_b M=\log_b(M^k)$
$\log_b M+\log_b N=\log_b(MN)$
$\log_b M-\log_b N=\log_b\left(\frac{M}{N}\right)$

The logarithms must have the same base, and every argument must be positive. Also, a sum of logarithms becomes the logarithm of a product, not a sum.

Example

Condense the expression:

2\log_3 x+\log_3(x-1)-\log_3 5

Assume $x>1$ , so both $x$ and $x-1$ are positive.

Step 1: Move the coefficient into the first logarithm as an exponent.

2\log_3 x=\log_3(x^2)

Now the expression is

\log_3(x^2)+\log_3(x-1)-\log_3 5

Step 2: Combine the sum using the product property.

\log_3(x^2)+\log_3(x-1)=\log_3\bigl(x^2(x-1)\bigr)

So we have

\log_3\bigl(x^2(x-1)\bigr)-\log_3 5

Step 3: Combine the difference using the quotient property.

\boxed{\log_3\left(\frac{x^2(x-1)}{5}\right)}

Thus,

2\log_3 x+\log_3(x-1)-\log_3 5 =\log_3\left(\frac{x^2(x-1)}{5}\right).

Learn by doing: Condense logarithmic expressions

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Logarithms - Product Property - Sum to Product as Integer

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