solve 5^(p+4)= 2^(4-5p)to 2 decimal places
take log of both sides
log(5^(p+4)) = log(2^(4-5p))
(p+4)log5 = (4-5p)log2
p log5 + 4log5 = 4log2 - 5p log2
p(log5 + 5log2) = 4log2 - 4log5
p = (4log2 - 4log5)/( log5 + 5log2) = appr -.72
To solve the equation 5^(p+4) = 2^(4-5p) to 2 decimal places, we need to isolate the variable p.
Step 1: Take the natural logarithm (ln) on both sides of the equation to eliminate the exponential terms:
ln(5^(p+4)) = ln(2^(4-5p))
Step 2: Apply the logarithmic properties to simplify the equation:
(p+4)ln(5) = (4-5p)ln(2)
Step 3: Distribute the natural logarithm values:
p * ln(5) + 4 * ln(5) = 4 * ln(2) - 5p * ln(2)
Step 4: Gather the like terms:
p * ln(5) + 5p * ln(2) = 4 * ln(2) - 4 * ln(5)
Step 5: Combine the "p" terms:
p(ln(5) + 5ln(2)) = 4ln(2) - 4ln(5)
Step 6: Divide both sides of the equation by (ln(5) + 5ln(2)):
p = (4ln(2) - 4ln(5)) / (ln(5) + 5ln(2))
Step 7: Use a calculator to evaluate the expression on the right-hand side to 2 decimal places:
p ≈ -0.0535
Therefore, the solution to the equation 5^(p+4) = 2^(4-5p) to 2 decimal places is p ≈ -0.0535.