Write the expression in a single natural logarithm. 3 ln a - 0.5 (ln b + ln c^2)
3 ln a - 0.5 (ln b + ln c^2)
ln a^3 - ln b^.5 - ln c
or
ln (a^3 /(c√b) )
To write the given expression in a single natural logarithm, we can simplify the expression using logarithmic properties.
The first step is to distribute the factor of 0.5 to the terms inside the parentheses:
3 ln a - 0.5 (ln b + ln c^2)
= 3 ln a - 0.5 ln b - 0.5 ln c^2
Next, we can use the property of logarithms that states ln(a) + ln(b) = ln(a * b):
3 ln a - 0.5 ln b - 0.5 ln c^2
= ln a^3 - ln b^0.5 - ln (c^2)^0.5
= ln a^3 - ln b^0.5 - ln c
Then, using the property that ln(a) - ln(b) = ln(a / b), we can combine the first two terms:
ln a^3 - ln b^0.5 - ln c
= ln (a^3 / b^0.5) - ln c
Finally, using the property that ln(a) - ln(b) = ln(a / b), we can combine the last two terms:
ln (a^3 / b^0.5) - ln c
= ln ((a^3 / b^0.5) / c)
Therefore, the expression can be written as a single natural logarithm: ln ((a^3 / b^0.5) / c).