Question

If ae^(ty)=c, where a,b, and c are positive, then t equals 1ln((c)/(ab)) (ln((c)/(a)))/(b) 2ln((cb)/(a)) (ln((c)/(a)))/(lnb)

Answer 1

To solve t from the equation ae^(ty)=c, you divide both sides by a, take the natural logarithm of both sides, and then divide by y. This yields t = (ln(c/a))/y.

Step-by-step explanation:

If aety=c, where a, c are positive constants, and y is a variable, to solve for t, we first isolate the exponent by dividing both sides of the equation by a:

ety = c/a

Next, we take the natural logarithm of both sides:

ln(ety) = ln(c/a)

Using the property that the natural logarithm of an exponential function with the same base simplifies to the exponent:

ty = ln(c/a)

Finally, we divide both sides by y to solve for t:

t = (ln(c/a))/y

This equation suggests that the value of t is obtained by dividing the natural logarithm of the ratio of c to a by the variable y. This process leverages the fundamental relationship between exponents and logarithms, particularly their ability to transform multiplicative relationships into additive ones.