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How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed?
$t=703,800,000\times \frac{\mathrm{ln}(0.8)}{\mathrm{ln}(0.5)}\text{years}\approx \text{}226,572,993\text{years}.$
Access these online resources for additional instruction and practice with exponential and logarithmic equations.
One-to-one property for exponential functions | For any algebraic expressions
$\text{}S\text{}$ and
$\text{}T\text{}$ and any positive real number
$\text{}b,\text{}$ where
${b}^{S}={b}^{T}\text{}$ if and only if $\text{}S=T.$ |
Definition of a logarithm | For any algebraic expression
S and positive real numbers
$\text{}b\text{}$ and
$\text{}c,\text{}$ where
$\text{}b\ne 1,$
${\mathrm{log}}_{b}(S)=c\text{}$ if and only if $\text{}{b}^{c}=S.$ |
One-to-one property for logarithmic functions | For any algebraic expressions
S and
T and any positive real number
$\text{}b,\text{}$ where
$\text{}b\ne 1,$
${\mathrm{log}}_{b}S={\mathrm{log}}_{b}T\text{}$ if and only if $\text{}S=T.$ |
How can an exponential equation be solved?
Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.
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