Given that,
so,

*using that find the value of
with out using logarithmic table or calculator. need step by step explanation.
will give the brainliest.

Step-by-step explanation:

We have:

$log_{10} (10)=1$

$\therefore \log_{10}(2\times 5)=1$

$\implies \log_{10}(2)+ \log_{10}(5)=1$

$\implies \log_{10}(5)=1-\log_{10}(2)$

$\implies \log_{10}(5)=1-0.3010=\boxed{0.6990}$

The answers to the question are

(a) The time series plot is given as attached

(b) The parameters for the line that minimizes MSE for the time series are;

y-intercept, b₀ = 19.993

Slope, b₁ =  1.77

MSE = 19.44

T = 19.993  + 1.774·t

(c) The average cost increase that the firm is realizing per year is $ 1.77

(d) The estimate of the cost/unit for next year is $35.96.

Step-by-step explanation:

(a) Using the provided data, the time series plot is given as attached

(b) Given hat the y-intercept, = b₀

Slope = b₁

Therefore the  linear trend forecast equation is given s

T = b₀ + b₁·t

The linear trend line slope is given as

b₁ =

b₀ = - b₁·

Where:

Y = Time series plot value at t

n =  Time period number

= Time series data average value and

= Average time, t

Therefore, =

= 4.5

 =  

 =  27.975

Therefore the linear trend line equation T, is

b₁ = =   = 1.774

b₀ = - b₁· = 27.975 - 1.774×4.5 = 19.993

Therefore the trend equation for the linear trend is

T = 19.993  + 1.774·t

MSE = = = 19.44

(c) From the linear trend equation, the average is given as the slope of the curve or b₁ which is equal to 1.774

Therefore the average cost increase that the firm has been realizing per year is $ 1.77

(b) From the equation of the future trend, we have when y = 9

T  is given as

T = 19.993  + 1.774×9 = 35.96

The cost/unit for 9th year is $35.96