Question

Solve 3 ^ (x - 8) = 8 for x using the change of base formula logb y = (log y)/(log b)

Answer 1

Answer:

$x=3\log_3 (2) +8$

Step-by-step explanation:

Here we are using power rule first.

Power rule = The logarithm of an exponential number is the exponent times the logarithm of the base $[log(a)^{b}=b\times log(a)]$.

For the function given.

$3^{(x-8)} = 8$,using log function both sides.

$(x-8)log(3)=log(8)$

Now,

$(x-8)=\frac{log(8)}{log(3)}$

Adding $8$ both sides.

$x=\frac{log(8)}{log(2)} +8$

And we know that $8=2^{3}$ so we can further write $log(8)=log(2^{3})=3log(2)$

Then we have $x=\frac{3\log(2)}{\log 3} +8$.

Now, using change of base formula:

$\frac{\log y}{\log b}=\log_b y$

So, $\frac{\log 2}{\log 3}=\log_3 2$

Our final answer is  $x=3\log_3 (2) +8$.

Answer 2

Answer:

x = 3 (Log 2 base 3) + 8

Step-by-step explanation:

3^(x - 8) = 8

Take the log of both side

Log 3^(x - 8) = Log 8

Recall:

log a^b = b log a

Log 3^(x - 8) = Log 8

(x - 8) Log 3 = Log 8

Divide both side by log 3

x - 8 = Log 8 / log 3

Recall:

8 = 2^3

x - 8 = Log 8 / log 3

x - 8 = Log 2^3 / log 3

x - 8 = 3 Log 2 / log 3

Recall:

logb y = log y / log b

log y / log b = logb y

x - 8 = 3 Log 2 / log 3

x - 8 = 3 Log3 2

Add 8 to both side

x = 3 Log3 2 + 8

x = 3 (Log 2 base 3) + 8