Question

(1.05)^x = 2.5

Round your answer to the nearest tenth. how would i do this step by step?

2.
___
√ 3x = 50 Round your answer to the nearest tenth.

Answer 1

Apply logarithm of base 10 to both the sides
\[\log_{10}((1.05)^{x})=\log_{10}(2.5)\]

Rule:
1.\[\log(x^y)=y.\log(x)\] (true for all base)
2.\[\log_{a}a=1\]
log of any number to it's base is 1
3.\[\log(a \times b)=\log(a)+\log(b)\]
4.\[\log(\frac{a}{b})=\log(a)-\log(b)\]
Also true for all base
5.\[\log(1)=0\]
Also true for all base
\[x.\log_{10}(1.05)=\log_{10}(\frac{10}{4})\]
Here's the first I've applied property 1 to left side and written right side 2.5 as 10/4 use property 4 on right side now
Note on the right side I've written 2.5 as 10/4

Answer 2

Ok so i just do (1.05)(10/4) or... what

Answer 3

Do you know about logarithms?

Answer 4

ok
Consider an example
\[(0.4)^x=\frac{0.25}{100}\]
Apply log of base 10 to both sides
\[\log_{10}(0.4)^x=\log_{10}(\frac{0.25}{100})\]
Use rule 1 on left side and rule 4 on right side
\[x.\log_{10}(0.4)=\log_{10}(0.25)-\log_{10}(100)\]
Now we can write 0.4 as 4/10 0.25 as square of 0.5 and 100 as square of 10 so do that
\[x.\log_{10}(\frac{4}{10})=\log_{10}((0.5)^2)-\log_{10}(10^2)\]
Property 4 on left and property property 1 on right
\[x[\log_{10}(4)-\log_{10}(10)]=2\log_{10}(0.5)-2\log_{10}(10)\]
We can write 4 as 2 square and 0.5 as 1/2 so do that
\[x[\log_{10}(2^2)-\log_{10}(10)]=2[\log_{10}(\frac{1}{2})-\log_{10}(10)]\]
Use properties to simplify further
\[x[2\log_{10}(2)-\log_{10}(10)]=2[\log_{10}(1)-\log_{10}(2)-\log_{10}(10)]\]
Now put the values in,
\[\log_{10}(2)=0.3010\]
By property 2 and property 5 the other values are
\[\log_{10}(10)=1\]\[\log_{10}(1)=0\]\[x[2 \times 0.3010-1]=2[0-0.3010-1]\]\[-0.398x=-2.06\]\[x=\frac{2.06}{0.398}\approx 5.176\]

Answer 5

sorry didnt mean to bump