Solve the exponential equation using logarithms. Give the answer in decimal form, rounding to four decimal places. (Enter your answers as a comma-separated list.)
4x − 3 = 32x
x = _____
I assume you meant
4^(x-3) = 32^x
Since 4=2^2 and 32=2^5, we have
2^(2x-6) = 2^(5x)
2x-6 = 5x
3x = -6
x = -2
Check: 4^(-5) = 2^-10
32^(-2) = 2^-10
no it looks more like this:
4^(x-3)=3^2x
To solve the exponential equation 4x - 3 = 32x using logarithms, we can take the logarithm of both sides with the same base. In this case, let's use the common logarithm (base 10) or the logarithm with base 10 (log₁₀).
1. Take logarithm (base 10) of both sides:
log₁₀(4x - 3) = log₁₀(32x)
2. Apply the properties of logarithms:
log₁₀(4x - 3) = log₁₀(32) + log₁₀(x)
3. Simplify the logarithms:
log₁₀(4x - 3) = 1.5051 + log₁₀(x)
4. Let's assume log₁₀(x) = a for simplicity:
log₁₀(4x - 3) = 1.5051 + a
5. Rewrite the equation without logarithms:
10^(log₁₀(4x - 3)) = 10^(1.5051 + a)
6. By definition, logarithm and exponential functions are inverses.
This simplifies the equation to:
4x - 3 = 10^(1.5051) * 10^a
7. Simplify the right side:
4x - 3 = 31.623 * 10^a
8. Rearrange the equation:
4x = 31.623 * 10^a + 3
9. Divide by 4:
x = (31.623 * 10^a + 3) / 4
Thus, the solution to the exponential equation 4x - 3 = 32x in decimal form, rounded to four decimal places, is x = (31.623 * 10^a + 3) / 4.
Please note that the value of 'a' will depend on the particular logarithm you choose for step 4.
To solve the exponential equation 4x - 3 = 32x using logarithms, we need to bring all the terms with the variable x to one side of the equation.
Step 1: Rewrite the equation to isolate the exponentials term:
4x = 32x + 3
Step 2: Move the like terms to one side of the equation:
4x - 32x = 3
Step 3: Combine the x terms:
-28x = 3
Step 4: Divide both sides by the coefficient of x:
-28x / -28 = 3 / -28
x = -3 / 28
Since the question asks for the answer in decimal form, rounding to four decimal places, we can calculate:
x ≈ -3 / 28 ≈ -0.1071
Therefore, the solution to the equation 4x - 3 = 32x is x ≈ -0.1071.