solve c---Solve for $x$, correct to 3 significant figures:
a $5^x = 40$
b $3^x = 2^{x+3}$
c $2^{x+4} = 5^{2-x}$
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Today we'll solve three exponential equations. We have 5 to the power x equals 40, 3 to the power x equals 2 to the power x plus 3, and 2 to the power x plus 4 equals 5 to the power 2 minus x. Let's focus on the third equation, which demonstrates the logarithmic method for solving exponential equations.
The first step in solving this exponential equation is to take the natural logarithm of both sides. This transforms the equation from 2 to the power x plus 4 equals 5 to the power 2 minus x, into ln of 2 to the power x plus 4 equals ln of 5 to the power 2 minus x. Using the logarithm property that ln of a to the power b equals b times ln of a, we can bring the exponents down to get x plus 4 times ln of 2 equals 2 minus x times ln of 5.
Now we expand both sides of the equation. On the left side, we distribute to get x times ln of 2 plus 4 times ln of 2. On the right side, we get 2 times ln of 5 minus x times ln of 5. Next, we collect all terms containing x on the left side and move the constants to the right side. This gives us x times ln of 2 plus x times ln of 5 equals 2 times ln of 5 minus 4 times ln of 2. Finally, we factor out x from the left side.
Now we can simplify the left side using the logarithm property that ln of a plus ln of b equals ln of a times b. Since ln of 2 plus ln of 5 equals ln of 2 times 5, which is ln of 10, our equation becomes x times ln of 10 equals 2 ln of 5 minus 4 ln of 2. To isolate x, we divide both sides by ln of 10, giving us x equals the fraction 2 ln of 5 minus 4 ln of 2, all over ln of 10.
Now we calculate the numerical value. We substitute ln of 5 approximately equals 1.6094, ln of 2 approximately equals 0.6931, and ln of 10 approximately equals 2.3026. This gives us 2 times 1.6094 minus 4 times 0.6931, all divided by 2.3026. Calculating step by step: 3.2188 minus 2.7724 equals 0.4464. Then 0.4464 divided by 2.3026 equals approximately 0.19386. Rounding to 3 significant figures, our final answer is x approximately equals 0.194.