Solve for x in the equation :(log_4x)²=log_16x^5+6
To solve the equation (log_4x)² = log_16x^5+6, we can first apply the properties of logarithms.
Using the logarithmic identity that states log_b(a^c) = c*log_b(a), we can rewrite the equation as:
2*log_4(x) = log_16(x^5+6)
Next, we can use the change of base formula to convert the logarithms to a common base. Let's choose base 2:
log_2(x) / log_2(4) = log_2(x^5+6) / log_2(16)
Simplifying further, we know that log_2(4) = 2 and log_2(16) = 4:
log_2(x) / 2 = log_2(x^5+6) / 4
To eliminate the fractions, we can cross multiply:
4*log_2(x) = 2*log_2(x^5+6)
Now, we can eliminate the logs again by applying the logarithmic identity log_b(a^c) = c*log_b(a):
log_2(x^4) = log_2((x^5+6)^2)
Since the bases are the same, the arguments must be equal:
x^4 = (x^5+6)^2
Expanding the right side of the equation:
x^4 = x^10 + 12x^5 + 36
Moving all terms to one side:
x^10 + 12x^5 + 36 - x^4 = 0
This is a degree 10 polynomial equation in terms of x. To obtain an exact solution, a numerical approximation method or the use of a computer algebra system may be necessary. Since the equation is quite complex, it is not straightforward to find an exact algebraic solution.
To solve the equation (log_4x)² = log_16x^5 + 6, we can break it down into smaller steps. Here's the step-by-step solution:
Step 1: Simplify the equation by using logarithmic properties
(log_4x)² = log_16x^5 + 6
Step 2: Apply the power rule of logarithms
2(log_4x) = log_16x^5 + 6
Step 3: Rewrite the right side of the equation using logarithmic properties
2(log_4x) = log_16 + log(x^5) + 6
Step 4: Simplify the logarithms using the power rule
2(log_4x) = 2 + 5log(x) + 6
Step 5: Simplify the equation by combining like terms
2log_4x - 5log(x) = 8
Step 6: Convert the logarithms to a common base (such as base 10)
(log(x) / log(4))^2 - 5log(x) = 8
Step 7: Solve for log(x)
Let's substitute log(x) with a variable, such as y.
(log(y) / log(4))^2 - 5y = 8
Step 8: Rewrite the equation as a quadratic equation by multiplying through by (log(4))^2
(log(y))^2 - 5(log(4))^2*y - 8*(log(4))^2 = 0
Now, the equation is in the form of a quadratic equation: ax^2 + bx + c = 0, where a = 1, b = -5(log(4))^2, and c = -8(log(4))^2.
Step 9: Solve the quadratic equation
At this point, you can use the quadratic formula to solve for y:
y = (-b ± sqrt(b^2 - 4ac)) / (2a)
Substituting the values, we get:
y = (5(log(4))^2 ± sqrt((5(log(4))^2)^2 - 4 * 1 * -8(log(4))^2)) / (2 * 1)
Simplifying further:
y = (5(log(4))^2 ± sqrt(25(log(4))^4 + 32(log(4))^2)) / 2
Please note that we have obtained the value of y, which represents log(x). To find the value of x, we need to raise 4 to the power of y.