Solve for x
a) e^2x+1=74
b) Log3x=4 (3 is subscript)
e^2x + 1 = 74
e^2x = 73
2x = ln73
x = ln73/2
log3(x) = 4
x = 3^4 = 81
a) To solve the equation e^(2x+1) = 74 for x, we need to isolate the variable x. Here's how to do it step by step:
Step 1: Take the natural logarithm (ln) on both sides of the equation to eliminate the exponential term:
ln(e^(2x+1)) = ln(74)
Step 2: Apply the logarithm property which states that ln(e^a) = a:
(2x+1) ln(e) = ln(74)
Step 3: Since ln(e) equals 1, we can simplify the equation:
2x + 1 = ln(74)
Step 4: Subtract 1 from both sides of the equation to isolate the term with x:
2x = ln(74) - 1
Step 5: Divide both sides by 2 to solve for x:
x = (ln(74) - 1) / 2
Hence, x = (ln(74) - 1) / 2 is the solution to the equation e^(2x+1) = 74.
b) To solve the equation log₃(x) = 4, we need to isolate x. Here's the step-by-step process:
Step 1: Rewrite the equation with exponential notation. The logarithm equation can be expressed as:
3^4 = x
Step 2: Simplify the left side of the equation:
81 = x
Hence, x = 81 is the solution to the logarithmic equation log₃(x) = 4.