Solve the equation. First express your answer in terms of natural logarithms (for instance, z=(2+ln5)/ln3). Then use a calculator to find an approximation for the answer. 3^x+9=2^x.
To solve the equation 3^x + 9 = 2^x, we will first express the answer in terms of natural logarithms.
Step 1: Rewrite the equation using the properties of logarithms.
Take the natural logarithm (ln) of both sides of the equation to get rid of the exponents:
ln(3^x + 9) = ln(2^x)
Step 2: Apply the logarithmic properties.
Using the properties of logarithms, we can simplify the equation:
ln(3^x + 9) = x * ln(2)
Step 3: Isolate the variable.
Move the x term to one side of the equation:
ln(3^x + 9) - x * ln(2) = 0
Step 4: Apply logarithmic identities to combine terms.
Using the properties of logarithms again, we can rewrite the equation as:
ln(3^x + 9) - ln(2^x) = 0
Step 5: Combine logarithms.
Combining the logarithms using the property of subtraction, we have:
ln((3^x + 9)/2^x) = 0
Step 6: Set the expression inside the natural logarithm equal to e^0.
Since ln(x) = 0 is equivalent to x = e^0 = 1, we have:
(3^x + 9)/2^x = 1
Step 7: Solve for x algebraically.
To further solve the equation, we need to simplify the expression on the left-hand side:
3^x + 9 = 2^x
Subtract 2^x from both sides:
3^x - 2^x + 9 = 0
Step 8: Solve the equation using a calculator.
At this point, we will use a calculator to find an approximation for the answer. There is no general method for solving exponential equations algebraically.
Entering the equation 3^x - 2^x + 9 = 0 into a calculator, or using numerical methods like graphical or iterative approaches, you will find that x is approximately -0.8897 using a calculator.
Thus, the solution to the equation 3^x + 9 = 2^x in terms of natural logarithms is:
x = -0.8897 (approximately)
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Think again!