# Solve for n 4*64^n=1 4⋅64n=1
Simplify.
Rewrite 4 as 22.
22⋅64n=1
Rewrite 64 as 26.
22⋅(26)n=1
Apply the power rule and multiply exponents, (am)n=amn.
22⋅26n=1
Use the power rule aman=am+n to combine exponents.
22+6n=1
22+6n=1
Divide each term in 22+6n=1 by 4.
22+6n4=14
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
22+6n=1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
ln(22+6n)=ln(1)
Expand ln(22+6n) by moving 2+6n outside the logarithm.
(2+6n)ln(2)=ln(1)
Apply the distributive property.
2ln(2)+6nln(2)=ln(1)
The natural logarithm of 1 is 0.
2ln(2)+6nln(2)=0
Subtract 2ln(2) from both sides of the equation.
6nln(2)=-2ln(2)
Divide each term by 6ln(2) and simplify.
Divide each term in 6nln(2)=-2ln(2) by 6ln(2).
6nln(2)6ln(2)=-2ln(2)6ln(2)
Simplify 6nln(2)6ln(2).
Cancel the common factor of 6.
Cancel the common factor.
6nln(2)6ln(2)=-2ln(2)6ln(2)
Rewrite the expression.
nln(2)ln(2)=-2ln(2)6ln(2)
nln(2)ln(2)=-2ln(2)6ln(2)
Cancel the common factor of ln(2).
Cancel the common factor.
nln(2)ln(2)=-2ln(2)6ln(2)
Divide n by 1.
n=-2ln(2)6ln(2)
n=-2ln(2)6ln(2)
n=-2ln(2)6ln(2)
Simplify -2ln(2)6ln(2).
Cancel the common factor of -2 and 6.
Factor 2 out of -2ln(2).
n=2(-ln(2))6ln(2)
Cancel the common factors.
Factor 2 out of 6ln(2).
n=2(-ln(2))2(3ln(2))
Cancel the common factor.
n=2(-ln(2))2(3ln(2))
Rewrite the expression.
n=-ln(2)3ln(2)
n=-ln(2)3ln(2)
n=-ln(2)3ln(2)
Cancel the common factor of ln(2).
Cancel the common factor.
n=-ln(2)3ln(2)
Rewrite the expression.
n=-13
n=-13
Move the negative in front of the fraction.
n=-13
n=-13
n=-13
The result can be shown in multiple forms.
Exact Form:
n=-13
Decimal Form:
n=-0.3‾
Solve for n 4*64^n=1

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