how do you solve .5=14^x?

Question

.5=14^x how do you solve this..im doign exponential growth, with the equation being y=(20)(.7)^x. Y=Number of miligrams, x=Hours.

How many hours would it take for the medicine dosage to go below .5 miligrams?

SO from y=(20)(.7)^x.
I got .5=(14)^x

Answer?

Answer 1

log14 .5 = x

So x = log .5/log 14

Answer 2

JimBurnell correctly told you how to solve .5 = 14^x

I urge you to double-check (20)(.7)^x = 14^x, though. If you plug in Jim's solution for .5 = 14^x, it works great, but it will fail when you plug it into y=(20)(.7)^x. y = 21.96, instead of .5.

The reason for this is that you cannot multiply bases of exponents together. You go from (20)(.7)^x = 14^x.
This is true for x = 1, but it fails for all other numbers.

To solve the problem, you need to go from:
y = (20)(.7)^x to:
y/20 = (.7)^x

When you plug in .5 for y, you get
(.7)^x = .025
x = log .025 / log .7

Answer 3

Actually you can't multiply 20 * 0.7^x and get 14^x since that isn't equivalent.

Start with:
0.5 = 20 * 0.7^x
Divide both sides by 20:
0.025 = 0.7^x

Take the log[0.7] of both sides:
log[0.7](0.025) = x

So how do you take log base 0.7? You need to convert bases:
If you have log[b](n) where b is the base and n is the number, then you can change bases by taking log(n) / log(b).

So using the formula to change bases you have:
x = log(0.025) / log(0.7)
x ≈ -1.602 / -0.155
x ≈ 10.34 hours

Answer 4

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