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 answer

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
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