Question

log base root 3 to the 243​

Answer 1

Step-by-step explanation:

$log_{\sqrt3} \: 243 \\ \\ = log_{\sqrt3} \: {3}^{5} \\ \\ = 5log_{\sqrt3} \: {3} \\ \\ = 5 \times \frac{log{3}}{log{\sqrt3}} \\ \\ = 5 \times \frac{log{3}}{log{ {3}^{ \frac{1}{2} } }} \\ \\ = 5 \times \frac{log{3}}{\frac{1}{2} log{ {3}}} \\ \\ = 5 \times \frac{2 \times log{3}}{ log{ {3}}} \\ \\ = 5 \times 2 \\ \\ = 10 \\ \\ \huge \purple{ \boxed{\therefore \: log_{\sqrt3} \: 243 = 10}}$

Answer 2

The value of the given logarithmic expression is 10.

Step-by-step explanation:

The given expression is

$\log_{\sqrt3}243$

Write 243 as the 243 = 3^5

$\log_{\sqrt3}3^5$

Now, write 3 = (√3)²

$\log_{\sqrt3}((\sqrt3)^2)^5$

Use the exponent property $(x^m)^n=x^{mn}$

$\log_{\sqrt3}((\sqrt3)^{10}$

Apply the log property $\log x^y=y\log x$

$10\log_{\sqrt3}((\sqrt3)$

Finally apply the log property $\log_a a=1$

$10\cdot1\\\\=10$

Thus, the value of the expression is 10.

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