Why Does Raising Both Sides of a Logarithmic Equation Yield a Different Result?

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Raising both sides of a logarithmic equation involves applying the properties of exponents correctly. The equation logS = a - 0.0018t transforms to S = 10^(a - 0.0018t) when both sides are raised to the power of 10. This is because the rule states that 10^(a - b) equals 10^a divided by 10^b, not the sum of the individual exponentials. Misunderstanding this can lead to incorrect conclusions, such as assuming 10^(a - b) equals 10^a - 10^b. Understanding these exponent rules is crucial for correctly manipulating logarithmic equations.
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Hi I don't understand why this is true:

If you have the function logS = a -0,0018t
and you raise both sides of the equation in 10 you should get

S = 10^(a) - 10^(-0,0018t)

but in my book they get

S= 10^(a - 0,0018t)

When you raise both sides of the equation in 10, should you not raise the individual terms on each side and NOT the whole side?
 
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christian0710 said:
Hi I don't understand why this is true:

If you have the function logS = a -0,0018t
and you raise both sides of the equation in 10 you should get

S = 10^(a) - 10^(-0,0018t)
No, you shouldn't! 10^(a+ b)= (10^a)(10^b) NOT "10^a+ 10^b".

but in my book they get

S= 10^(a - 0,0018t)

When you raise both sides of the equation in 10, should you not raise the individual terms on each side and NOT the whole side?
You "should" learn the rules of of exponents:
x^(a+ b)= (x^a)(x^b)
and
(x^a)^b= x^(ab)
 
christian0710 said:
Hi I don't understand why this is true:

If you have the function logS = a -0,0018t
and you raise both sides of the equation in 10 you should get

S = 10^(a) - 10^(-0,0018t)

but in my book they get

S= 10^(a - 0,0018t)

When you raise both sides of the equation in 10, should you not raise the individual terms on each side and NOT the whole side?

You seem to think that 10^(a-b) = 10^a - 10^b. Why don't you check this out for yourself? If a = 2 and b = 1, we have c = a-b = 2-1 = 1, so 10^c = 10^1 = 10. Do you agree so far? Now 10^a - 10^b = 10^2 - 10^1 = 100 - 10 = 90. OK still? So, now: do you really think that 10 = 90?

In general, what is true is that ##10^{a+b} = 10^a \times 10^b## and ##10^{a-b} = 10^a \div 10^b##. In fact, that is the whole point of logarithms: you can do multiplication or division by addion or subtraction of logarithms.
 
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