Answer :

We start with the equation

[tex]$$
0.27^{\,x-6}=9^{\,x-7}.
$$[/tex]

Taking the natural logarithm of both sides, we have

[tex]$$
\ln\left(0.27^{\,x-6}\right)=\ln\left(9^{\,x-7}\right).
$$[/tex]

Using the logarithm power rule, [tex]$\ln(a^b)=b\ln(a)$[/tex], the equation becomes

[tex]$$
(x-6)\ln(0.27) = (x-7)\ln(9).
$$[/tex]

Expanding both sides gives

[tex]$$
x\ln(0.27) - 6\ln(0.27) = x\ln(9) - 7\ln(9).
$$[/tex]

Next, we gather the terms involving [tex]$x$[/tex] on one side and the constant terms on the other side:

[tex]$$
x\ln(0.27) - x\ln(9) = 6\ln(0.27) - 7\ln(9).
$$[/tex]

Factor [tex]$x$[/tex] out of the left-hand side:

[tex]$$
x\Bigl[\ln(0.27)-\ln(9)\Bigr] = 6\ln(0.27)-7\ln(9).
$$[/tex]

Now, solve for [tex]$x$[/tex] by dividing both sides by [tex]$\ln(0.27)-\ln(9)$[/tex]:

[tex]$$
x=\frac{6\ln(0.27)-7\ln(9)}{\ln(0.27)-\ln(9)}.
$$[/tex]

Plugging in the numerical approximations

[tex]$$
\ln(0.27)\approx -1.3093333199837622 \quad \text{and} \quad \ln(9)\approx 2.1972245773362196,
$$[/tex]

first compute the numerator:

[tex]$$
6\ln(0.27)-7\ln(9)\approx 6(-1.3093333199837622)-7(2.1972245773362196)\approx -23.23657196125611.
$$[/tex]

Then, compute the denominator:

[tex]$$
\ln(0.27)-\ln(9) \approx -1.3093333199837622-2.1972245773362196 \approx -3.506557897319982.
$$[/tex]

Dividing the numerator by the denominator gives

[tex]$$
x\approx\frac{-23.23657196125611}{-3.506557897319982}\approx 6.626604391450525.
$$[/tex]

Therefore, the solution to the equation is

[tex]$$
\boxed{x\approx 6.63.}
$$[/tex]

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