1. Write the given expression as a single logarithm.
log(70x)+log(10y)-2
2. Write the given expression as a single logarithm.
ln7e/(square root 2x)-ln(square root 2ex)
3. Write the given expression as a single logarithm.
-2Ln(x)+Ln(4y)-Ln(5z)
1. log (70x/(10y)) - 2
= log (7x/y) - 2
2. not clear if the √(2x) is part of the first ln or not.
3. -2lnx + ln(4y) - ln(5z)
= ln [ (x^2)(4y)/(5z) ]
To write the given expressions as a single logarithm, we can use the logarithmic properties, specifically the product, quotient, and power rules.
1. log(70x) + log(10y) - 2:
Using the product rule, we can combine the terms inside the logarithms:
log(70x * 10y) - 2
Simplifying, we get:
log(700xy) - 2
2. ln(7e)/(sqrt(2x)) - ln(sqrt(2ex)):
Using the quotient rule, we can divide the terms inside the logarithms:
ln(7e/(sqrt(2x)) / (sqrt(2ex)))
Simplifying the denominator, we get:
ln(7e/(sqrt(2x) * sqrt(2ex)))
Next, we can use the power rule to combine the terms inside the logarithm:
ln(7e/(sqrt(4x^2 * ex)))
Simplifying, we get:
ln(7e/(2x * sqrt(ex)))
Further simplifying, we get:
ln(7e / (2 * sqrt(x * ex)))
3. -2Ln(x) + Ln(4y) - Ln(5z):
Using the power rule, we can write the expression as:
Ln(x^(-2)) + Ln(4y) - Ln(5z)
Using the product rule, we can combine the terms inside the logarithms:
Ln((x^(-2)) * 4y) - Ln(5z)
Next, using the power rule, we get:
Ln((4y) / (x^2)) - Ln(5z)