# exponential equation Find the solution of the exponential equation 10^(1 - x) = 6^x

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We have to find the solution of the exponential equation 10^(1 - x) = 6^x

10^(1 - x) = 6^x

it is not possible to make the base equal, so logarithms can be used.

log[10^(1 - x)] = log[6^x]

=> (1 - x)log 10 = x*log 6

=> 1 - x = x*log 6

=> x( 1 + log 6) = 1

=> x = 1/(1 + log 6)

**The solution of the equation is x = 1/(1 + log 6)**

We can use logarithms to solve exponential equations.

We'll take the common logarthim both sides:

lg 10^(1 - x) = lg 6^x

We'll apply the power rule for logarithms:

(1-x) lg 10 = x lg 6

We'll recall that lg 10 = 1

We'll re-write the equation:

1 - x = x lg 6

We'll add x both sides:

x + xlg6 = 1

We'll factorize by x:

x(1 + lg 6) = 1

We'll divide by 1 + lg 6 = lg 10 + lg 6 = lg 60:

x = 1/lg 60

Rounded to four decimal places:

x = 0.5624