Solve each equation. Round decimal answers to the nearest thousandth.

5(0.10)^x = 4(0.12)^x

5(0.10)^x = 4(0.12)^x.

Take log of both sides:
log5 + xlog(0.10) = log4 + xlog(0.12)0.6990 - x = 0.6021 - 0.9208x,
-x + 0.9208x = 0.6021 - 0.6990,
-0.0792x = -0.0969,

X = -0.0969 / -0.0792 = 1.22.

CORRECTION:

Take log of both sides:
log5 + Xlog(0.10) = log4 + Xlog(0.12),
0.6990 - X = 0.6021 - 0.9208X,
-x + 0.9208X = 0.6021- 0.6990,
-0.0792X = -0.0969,

X = -0.0969 / -0.0792 = 1.22.

To solve the equation, we can start by dividing both sides of the equation by 4(0.12)^x:

(5(0.10)^x) / (4(0.12)^x) = 1

Next, we can simplify the expression in the numerator:

(5/4) * (0.10)^x / (0.12)^x = 1

Now, let's rewrite the equation using exponent properties and rearrange terms:

[(5/4) * (0.10 / 0.12)]^x = 1

Simplifying further:

[(5/4) * (1.20)]^x = 1

(6/4)^x = 1

Now, we can write the equation as an exponential equation:

(3/2)^x = 1

Since any number (except 0) raised to the power of 0 is equal to 1, we have:

x = 0

Therefore, the solution to the equation is x = 0.

To solve the equation 5(0.10)^x = 4(0.12)^x and round decimal answers to the nearest thousandth, follow these steps:

Step 1: Start by dividing both sides of the equation by (0.12)^x to isolate the variable on one side:
(5(0.10)^x) / (4(0.12)^x) = 1

Step 2: Simplify the expression by dividing 5/4:
(5/4) * [(0.10)^x / (0.12)^x] = 1

Step 3: To simplify further, combine the bases with the same exponent by using the rule (a^m / b^m) = (a/b)^m:
(5/4) * [(0.10/0.12)^x] = 1

Step 4: Calculate the division of 0.10/0.12:
(5/4) * (0.8333...)^x = 1

Step 5: The equation can be rewritten as:
(5/4) * (0.8333...)^x = 1^1

Step 6: Since any number raised to the power of 1 is equal to itself, the equation can be simplified as:
(5/4) * (0.8333...)^x = 1

Step 7: Multiply both sides of the equation by 4/5 to isolate the exponential term:
[(5/4) * (0.8333...)^x] * (4/5) = (1) * (4/5)

Step 8: Simplify:
(0.8333...)^x = 4/5

Step 9: To solve for x, take the logarithm (base 0.8333...) of both sides of the equation:
log base 0.8333... [(0.8333...)^x] = log base 0.8333... (4/5)

Step 10: Simplify:
x = log base 0.8333... (4/5)

Step 11: Use a calculator to evaluate log base 0.8333... (4/5) and round the decimal answer to the nearest thousandth.