how do I solve 60*3^(x/8) = 30*2(x/5)
I made a typo the equation is: 60*3^(x/8) = 30*2^(x/5)
60*3^(x/8) = 30*2^(x/5)
2 * 3^(x/8) = 2^(x/5)
log2 + x/8 log3 = x/5 log2
x (log2/5 - log3/8) = log2
x = log2/(log2/5 - log3/8) = 532
I would divide both sides by 30
2*3^(x/8) = 2^(x/5)
log both sides, and use basic log rules
log 2 + (x/8)log3 = (x/5)log2
x[ (1/5)log2 - (1/8)log3 ] = log2
now go to your calculator and evaluate, I got x = appr. 532.00
Your calculator will kick into scientific notation since the numbers are really big. If you have a half-decent calculator it should have more than one memory storages, use them to store in-between answers to get my answer.
Don't know what the purpose of solving such an equation serves,
the equation does not really represent anything measurably in
a real world.
actually, the original problem was
Two bacteria cultures are being studied in a lab. At the start, bacteria A had a population of 60 bacteria and the number of bacteria was tripling every 8 days. Bacteria B had a population of 30 bacteria and was doubling every 5 days. Determine the number of days it will take for both bacteria cultures to have the same population.
Sry guys for not responding I forgot I posted here.
I have already solved it.
To solve the equation 60 * 3^(x/8) = 30 * 2^(x/5), you need to use logarithms. Here's a step-by-step guide on how to solve it:
Step 1: Simplify the equation.
60 * 3^(x/8) = 30 * 2^(x/5)
Step 2: Divide both sides by 30 to simplify the equation further.
2 * 3^(x/8) = 2^(x/5)
Step 3: Take the logarithm of both sides. You can choose any logarithm base, but the common choice is the natural logarithm or base 10 logarithm.
ln(2 * 3^(x/8)) = ln(2^(x/5))
Step 4: Use properties of logarithms to simplify the equation.
ln(2) + ln(3^(x/8)) = (x/5) * ln(2)
Step 5: Expand the logarithm of 3^(x/8) using the exponential property of logarithms.
ln(2) + (x/8) * ln(3) = (x/5) * ln(2)
Step 6: Distribute the (x/5) to both terms on the right side.
ln(2) + (x/8) * ln(3) = (x/5) * ln(2)
Step 7: Move all terms involving x to one side of the equation.
(x/8) * ln(3) - (x/5) * ln(2) = -ln(2)
Step 8: Combine the terms involving x.
(x/8) * ln(3) * 5 - (x/5) * ln(2) * 8 = -ln(2)
Step 9: Multiply through by the least common denominator (40), getting rid of the fractions.
5x * ln(3) - 8x * ln(2) = -40 * ln(2)
Step 10: Factor out x.
x * (5 * ln(3) - 8 * ln(2)) = -40 * ln(2)
Step 11: Divide both sides by (5 * ln(3) - 8 * ln(2)).
x = (-40 * ln(2)) / (5 * ln(3) - 8 * ln(2))
Step 12: Use a calculator with logarithmic functions to approximate the value of x.