1. Write the expression as a single logarithm. log6 25 - log6 5
2. Solve the exponential equation for n. 25^2n+1 = 625
3. Solve the logarithmic equation. Round to the nearest hundredth if necessary. 2 log 6x = 3
4. Solve In(3x + 4) = 5. Round to the nearest thousandths.
5. Use Natural Logarithms to solve the equation. Round to the nearest thousandths. 3e^3x-5 = 39
1. log6 (25/5) = log6 5
2. 25^(2n+1) = 625
Since 625 = 5^4, we can rewrite the equation as:
(5^2)^(2n+1) = 5^4
5^(4n+2) = 5^4
Now, equating the exponents:
4n + 2 = 4
4n = 2
n = 1/2
3. 2 log6x = 3
log6x^2 = 3
x^2 = 6^3
x^2 = 216
x = ±√216
x ≈ ±14.696
4. ln(3x + 4) = 5
Taking the exponential of both sides:
e^(ln(3x + 4)) = e^5
3x + 4 = e^5
3x = e^5 - 4
x = (e^5 - 4)/3
x ≈ 57.216
5. 3e^(3x-5) = 39
Divide both sides by 3:
e^(3x-5) = 13
Take the natural logarithm of both sides:
ln(e^(3x-5)) = ln(13)
3x - 5 = ln(13)
3x = ln(13) + 5
x = (ln(13) + 5)/3