Question

(x+2)log_55^x = X How do I solve this?

Answer 1

\[(x+2)\log_{5}5^{x} = X \]

Answer 2

Are you sure on right side you have capital X ?

Answer 3

Same x on both sides

Answer 4

my bad

Answer 5

The power rule of logarithm says:

\[\log(x)^y = y \cdot \log(x)\]

Answer 6

\[(x+2)\log_{5}5^{x} = x \implies (x+2) \cdot x \cdot \log_5(5) = x\]

Getting?

Answer 7

bring it all to the front? why not to the exponent?

Answer 8

but yes I get it

Answer 9

Bring the exponent to the front in multiplication.. exponent is \(x\) there, so bring it front, okay?

Answer 10

okay

Answer 11

The other rule says:
If base and argument is same then value of logarithm is \(1\) ie:
\[\log_a(a) = 1\]

Answer 12

This way:

\(\log_5(5) = ??\)

Answer 13

so (x+2) * x * 1 = x?

Answer 14

Yep.. :)

Answer 15

so then x(x+2) = x?

Answer 16

Yes, distribute x to brackets...

Answer 17

x^2 + 2x = 2

Answer 18

2 or x??

Answer 19

\(x^2 + 2x = x\) Right?

Answer 20

\[x^2 = -2x\]

Answer 21

Wait, don't go fast..

Answer 22

oops I did something wrong haha

Answer 23

\(x^2 + 2x = x\)

Subtract \(x\) from both the sides.. :)

Answer 24

so x^2 + x = 0?

Answer 25

Yep, take \(x\) common from left hand side now.

Answer 26

x(x + 1) = 0

Answer 27

Great, well done.. :)

Answer 28

Now, can you find \(x\) ??

Answer 29

See, when this situation happens : \((x+a)(x+b) = 0\)

Then either (x+a) = 0 or (x+b) = 0.. Use this..

Answer 30

x = -1

Answer 31

and x = 0

Answer 32

Thats the answer in my book haha Thanks! :)

Answer 33

So the two answers are : \(x =0,1\)

Answer 34

\(\text{This is it}..\)