Solve each equation for x
e^(4x+1)=p
log (kx)/ log (5) = c
e^(4x+1)=p
ln(e^(4x+1)) = ln p
(4x+1) lne = lnp , but lne =1
4x+1 = lnp
4x = lnp - 1
x = (lnp - 1)/4
log (kx)/ log (5) = c
log( kx/log5) = c
kx/log5 = 10^c
kx = log5(10^c)
x = log5(10^c)/k
scrap the solution to the second equation, I misread the question
here is the correct solution:
log (kx)/ log (5) = c
log kx = c log5
kx = 10^(c log5)
x = 10^(c log5) / k
To solve the first equation, e^(4x+1) = p, we can take the natural logarithm (ln) on both sides of the equation. This allows us to eliminate the exponential function and convert it into a linear equation.
Step 1: Take the natural logarithm of both sides:
ln(e^(4x+1)) = ln(p)
Step 2: Simplify the left side using the property of logarithms:
(4x+1)ln(e) = ln(p)
Step 3: Simplify further:
4x+1 = ln(p)
Step 4: Solve for x by isolating the variable:
4x = ln(p) - 1
Step 5: Divide both sides by 4 to get the value of x:
x = (ln(p) - 1)/4
Therefore, the solution for the equation e^(4x+1) = p is x = (ln(p) - 1)/4.
To solve the second equation, log(kx) / log(5) = c, we can use the property of logarithms to change the base.
Step 1: Multiply both sides of the equation by log(5) to move the denominator:
log(kx) = c * log(5)
Step 2: Use the property of logarithms to change the base from log(5) to natural logarithm (ln):
log(kx) = ln(5) * c
Step 3: Rewrite the equation in exponential form:
kx = e^(ln(5) * c)
Step 4: Divide both sides of the equation by k to isolate the variable x:
x = (e^(ln(5) * c)) / k
Therefore, the solution for the equation log(kx) / log(5) = c is x = (e^(ln(5) * c)) / k.