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 the given values of logarithms, specifically log 3 = 0.4771.
Step 1: Rewrite the equation using logarithms
Take the logarithm of both sides of the equation using the base 3 because we have a base of 3 in the equation:
log3(3^(x+2)) = log3(405)
Step 2: Apply the power rule of logarithms
The power rule states that if we have logb(a^c), it can be written as c * logb(a). Applying this rule, we get:
(x+2) * log3(3) = log3(405)
Step 3: Simplify the left side of the equation
Since log3(3) equals 1, we can simplify further:
x+2 = log3(405)
Step 4: Evaluate log3(405)
To find the value of log3(405), we need to express 405 as a power of 3.
405 can be written as 3^4 * 5 because 405 / 81 = 5.
Therefore:
log3(3^4 * 5) = log3(3^4) + log3(5) = 4 * log3(3) + log3(5) = 4 + log3(5)
Step 5: Substitute the value of log3(405) into the equation
We substitute 4 + log3(5) into the previous equation:
x + 2 = 4 + log3(5)
Step 6: Solve for x
Rearrange the equation to isolate x:
x = (4 + log3(5)) - 2 = 4 - 2 + log3(5)
Step 7: Substitute the value of log3(5)
Using the given value log3(5) = 0.6990, substitute it into the equation:
x = 2 + 0.6990
Step 8: Calculate the numerical value of x
Add the numbers together:
x = 2 + 0.6990 = 2.699
Therefore, the numerical value of x in the equation 3^(x+2) = 405 is approximately 2.699.