If log 2 = .3010 log 3 = .4771, find the numerical value of x in 3^x+2 = 405
To find the numerical value of x in the equation 3^(x+2) = 405, we can use logarithmic properties.
Step 1:
Let's start by isolating the exponent on the left side of the equation. Divide both sides of the equation by 3^2 (since 3^2 = 9). This gives us:
3^(x+2) / 3^2 = 405 / 9
Simplifying this equation, we have:
3^(x+2-2) = 45
3^x = 45
Step 2:
Now, take the logarithm of both sides of the equation using any base. We can use the common logarithm (log base 10) or the natural logarithm (ln, log base e). Let's use the common logarithm (log base 10) in this case:
log (3^x) = log 45
Step 3:
Apply the exponent rule of logarithms, which states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number:
x log 3 = log 45
Step 4:
Now, substitute the given values of log 3 into the equation:
x * 0.4771 = log 45
Simplifying this equation, we have:
0.4771x = log 45
Step 5:
Divide both sides of the equation by 0.4771 to solve for x:
x = log 45 / 0.4771
Step 6:
Using a calculator, find the value of log 45:
log 45 ≈ 1.6532
Finally, divide 1.6532 by 0.4771:
x ≈ 1.6532 / 0.4771 ≈ 3.4612
Therefore, the numerical value of x in the equation 3^(x+2) = 405 is approximately 3.4612.