1) log9x + 2log3x = 5
Rewrite the logarithms using the power rule:
log(9x) + log(3x)^2 = 5
Combine the logarithms using the product and power rules:
log(9x) + log(9x)^2 = 5
Simplify the expression:
log(9x) + log(81x^2) = 5
Combine the logarithms using the product rule:
log(9x * 81x^2) = 5
log(729x^3) = 5
Convert to exponential form:
729x^3 = 10^5
729x^3 = 100000
x^3 = 100000 / 729
x^3 = 137.174
x = ∛137.174
x = 5.600
2) log2(x^2 - 3) + 1 = log2(6x - 10)
Subtract 1 from both sides:
log2(x^2 - 3) = log2(6x - 10) - 1
Use the properties of logarithms to combine the logarithms on the right side:
log2(x^2 - 3) = log2((6x - 10) / 2)
Since the bases are the same, the arguments must be equal:
x^2 - 3 = (6x - 10) / 2
Simplify the equation:
2x^2 - 6 = 6x - 10
Rearrange the equation to set it equal to zero:
2x^2 - 6x + 4 = 0
Solve the quadratic equation by factoring or using the quadratic formula:
(x - 2)(2x - 2) = 0
x = 2 or x = 1
Therefore, the solutions to the equation are x = 2 or x = 1.
1) log9x + 2log3x = 5
Rewrite the logarithms using the power rule:
log(9x) + log(3x)^2 = 5
Combine the logarithms using the product and power rules:
log(9x) + log(9x)^2 = 5
Simplify the expression:
log(9x) + log(81x^2) = 5
Combine the logarithms using the product rule:
log(9x * 81x^2) = 5
Simplify the expression:
log(729x^3) = 5
Convert to exponential form:
729x^3 = 10^5
729x^3 = 100000
x^3 = 100000 / 729
x^3 = 137.174
x = ∛137.174
x = 5.600
2) log2(x^2 - 3) + 1 = log2(6x - 10)
Subtract 1 from both sides:
log2(x^2 - 3) = log2(6x - 10) - 1
Use the properties of logarithms to combine the logarithms on the right side:
log2(x^2 - 3) = log2((6x - 10) / 2)
Since the bases are the same, the arguments must be equal:
x^2 - 3 = (6x - 10) / 2
Simplify the equation:
2x^2 - 6 = 6x - 10
Rearrange the equation to set it equal to zero:
2x^2 - 6x + 4 = 0
Solve the quadratic equation by factoring or using the quadratic formula:
(x - 2)(2x - 2) = 0
x = 2 or x = 1
Therefore, the solutions to the equation are x = 2 or x = 1.