Expand logarithmic expressions Expanding logarithmic expressions means rewriting logarithms of products, quotients, and positive powers as sums, differences, and coefficient multiples, using log b ( x y ) = log b x + log b y \log_b(xy)=\log_b x+\log_b y log b ( x y ) = log b x + log b y , log b ( x / y ) = log b x − log b y \log_b(x/y)=\log_b x-\log_b y log b ( x / y ) = log b x − log b y , and log b ( x r ) = r log b x \log_b(x^r)=r\log_b x log b ( x r ) = r log b x . The reasoning preserves the expression’s domain and distinguishes these valid transformations from the invalid distribution of logarithms over sums or differences; treatment is limited to real logarithms with b > 0 b>0 b > 0 , b ≠ 1 b\ne1 b = 1 , and positive arguments.
Detailed Explanation: Expand logarithmic expressions To expand a logarithmic expression, rewrite it using these rules:
log b ( x y ) = log b x + log b y \log_b(xy)=\log_b x+\log_b y log b ( x y ) = log b x + log b y log b ( x y ) = log b x − log b y \log_b\left(\frac{x}{y}\right)=\log_b x-\log_b y log b ( y x ) = log b x − log b y log b ( x r ) = r log b x \log_b(x^r)=r\log_b x log b ( x r ) = r log b x These rules apply to real logarithms when b > 0 b>0 b > 0 b ≠ 1 b\ne 1 b = 1
Example
Expand:
log 2 ( 8 x 3 y z 2 ) \log_2\left(\frac{8x^3\sqrt{y}}{z^2}\right) log 2 ( z 2 8 x 3 y ) Assume x > 0 x>0 x > 0 y > 0 y>0 y > 0 z > 0 z>0 z > 0
Step 1: Separate the quotient.
Use the quotient rule:
log 2 ( 8 x 3 y z 2 ) = log 2 ( 8 x 3 y ) − log 2 ( z 2 ) \log_2\left(\frac{8x^3\sqrt{y}}{z^2}\right)
=
\log_2(8x^3\sqrt{y})-\log_2(z^2) log 2 ( z 2 8 x 3 y ) = log 2 ( 8 x 3 y ) − log 2 ( z 2 ) Step 2: Expand the product in the numerator.
Use the product rule:
log 2 ( 8 x 3 y ) = log 2 8 + log 2 ( x 3 ) + log 2 ( y ) \log_2(8x^3\sqrt{y})
=
\log_2 8+\log_2(x^3)+\log_2(\sqrt{y}) log 2 ( 8 x 3 y ) = log 2 8 + log 2 ( x 3 ) + log 2 ( y ) So the expression becomes
log 2 8 + log 2 ( x 3 ) + log 2 ( y ) − log 2 ( z 2 ) \log_2 8+\log_2(x^3)+\log_2(\sqrt{y})-\log_2(z^2) log 2 8 + log 2 ( x 3 ) + log 2 ( y ) − log 2 ( z 2 ) Step 3: Move powers in front as coefficients.
Since y = y 1 / 2 \sqrt{y}=y^{1/2} y = y 1/2
log 2 ( x 3 ) = 3 log 2 x \log_2(x^3)=3\log_2 x log 2 ( x 3 ) = 3 log 2 x log 2 ( y ) = log 2 ( y 1 / 2 ) = 1 2 log 2 y \log_2(\sqrt{y})=\log_2(y^{1/2})=\frac12\log_2 y log 2 ( y ) = log 2 ( y 1/2 ) = 2 1 log 2 y and
log 2 ( z 2 ) = 2 log 2 z \log_2(z^2)=2\log_2 z log 2 ( z 2 ) = 2 log 2 z Also, log 2 8 = 3 \log_2 8=3 log 2 8 = 3 2 3 = 8 2^3=8 2 3 = 8
Therefore, the expanded expression is
3 + 3 log 2 x + 1 2 log 2 y − 2 log 2 z \boxed{3+3\log_2 x+\frac12\log_2 y-2\log_2 z} 3 + 3 log 2 x + 2 1 log 2 y − 2 log 2 z Remember that logarithms can be expanded over products and quotients, but not over sums or differences. For example,
log b ( x + y ) ≠ log b x + log b y \log_b(x+y)\ne \log_b x+\log_b y log b ( x + y ) = log b x + log b y