Lowest bound on logarthmic equation with floor

Question

I have the following equation (log base 10):

$$\frac{x}{10^{\lfloor \log x/10 \rfloor}}$$ how can I show what the maximum value of this expression can be? i.e. $\frac{x}{10^{\lfloor \log x/10 \rfloor}} < y$.

Answer 1

For your original question:

Let $n$ denote the number of decimal digits in the integer part of $x$:

• $x = d_{1} \dots d_{n}$
• $\lfloor\log{x}\rfloor = n-1$
• $10^{\lfloor\log{x}\rfloor} = 1\underbrace{0\dots0}_{n-1\text{ times}}$

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Therefore:

$$\frac{x}{10^{\lfloor\log{x}\rfloor}} = \frac{d_{1} \dots d_{n}}{1\underbrace{0\dots0}_{n-1\text{ times}}}<10^1$$

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For your updated question:

Let $n$ denote the number of decimal digits in the integer part of $x$:

• $x = d_{1} \dots d_{n}$
• $\lfloor\log{x}\rfloor = n-1$
• $\lfloor\log{x/10}\rfloor = n-2$
• $10^{\lfloor\log{x/10}\rfloor} = 1\underbrace{0\dots0}_{n-2\text{ times}}$

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Therefore:

$$\frac{x}{10^{\lfloor\log{x/10}\rfloor}} = \frac{d_{1} \dots d_{n}}{1\underbrace{0\dots0}_{n-2\text{ times}}}<10^2$$