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3 years ago

What is the value of x in the equation log5 x = 4 jmap?

1 answer:

5 0

To solve for x we proceed as follows:
from the laws of logarithm, given that:
log_a b=c
then
a^c=b
applying the rationale to our question we shall have:
log_5 x=4
hence
5^4=x
x=625
Answer: x=625

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3 years ago

2. Determine the sum of the first 400 ODD numbers.<br><br>

il63 [147K]

Odd numbers take the form $2n-1$ , where $n\ge1$ is an integer. When $n=400$ , the last odd number would be 799. So we're adding

$S=1+3+5+\cdots+795+797+799$

By reversing the order of terms, we have

$S=799+797+795+\cdots+5+3+1$

and we can pair up terms in both sums at the same position to write

$2S=(1+799)+(3+797)+(5+795)+\cdots(795+5)+(797+3)+(799+1)$

so that we are basically adding 400 copies of 800, and from there we can find the value of the sum right away:

$2S=400\cdot800\implies S=160,000$

###

We could also make use of the formulas,

$\displaystyle\sum_{i=1}^n1=n$

$\displaystyle\sum_{i=1}^ni=\dfrac{n(n+1)}2$

We have

$S=\displaystyle\sum_{i=1}^{400}(2i-1)=2\sum_{i=1}^{400}i-\sum_{i=1}^{400}1=400(400+1)-400=400^2=160,000$

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