Condensing Multiple Logarithms into a Single Log
Key Concepts
- Focus of this lesson is on condensing multiple logarithmic expressions into a single logarithm.
- Opposite process of expanding a single log into multiple logs.
- Important to have watched the previous lesson on expansion of logs.
Steps to Condense Logarithms
-
Identify Signs and Terms
- Terms with positive signs go in the numerator.
- Terms with negative signs go in the denominator.
-
Move Coefficients to Exponent Position
- Convert coefficients in front of the log into exponents.
Example 1
- Given:
2 log a + 5 log b - 7 log c
- Convert coefficients to exponents:
log a^2 + log b^5 - log c^7
- Combine into a single log:
- Numerator:
a^2 * b^5 (positive signs)
- Denominator:
c^7 (negative sign)
- Result:
log (a^2 * b^5 / c^7)
Example 2
- Given:
1/3 log a - 2/3 log b + 1/4 log c - 4/5 log d
- Move coefficients to exponents:
log a^(1/3) - log b^(2/3) + log c^(1/4) - log d^(4/5)
- Combine into a single log:
- Numerator:
a^(1/3) * c^(1/4) (positive signs)
- Denominator:
b^(2/3) * d^(4/5) (negative signs)
- Result:
log (a^(1/3) * c^(1/4) / (b^(2/3) * d^(4/5)))
- Optionally convert to radical form:
cube root of a * fourth root of c / (cube root of b^2 * fifth root of d^4)
Summary
- Coefficients are moved to exponents before combining logs.
- Positive terms in the numerator, negative terms in the denominator.
- Optional conversion to radical form for simplification.
By following these steps, you can condense any expression with multiple logarithms into a single logarithmic expression.